First, please post your initial R code you tried. I think you are talking about sample and matrix design of your experiment, for this please refer limma package manual.

look at some previous post:

Limma Multiple Group Analysis

Limma analysis for multiple groups

Limma Contrast matrix

Though in GSE24422 data there are total 24 samples, including replicate (12*2). So the sample Id is your group, you can define as follow:

sample code:
  group <- factor(rep(c("Mixed_rep", "Mixed_Ins_rep", "Mixed_TNF_rep", "Mixed_TNF_Ins_rep", "Adip_rep", "Adip_Ins_rep", "Adip_TNF_rep", "Adip_TNF_Ins_rep", "Strom_rep", "Strom_Ins_rep", "Strom_TNF_rep", "Strom_TNF_Ins_rep"),each=2))
 design <- model.matrix(~0 +group)
 colnames(design) <- c("Mixed_rep", "Mixed_Ins_rep", "Mixed_TNF_rep", "Mixed_TNF_Ins_rep", "Adip_rep", "Adip_Ins_rep", "Adip_TNF_rep", "Adip_TNF_Ins_rep", "Strom_rep", "Strom_Ins_rep", "Strom_TNF_rep", "Strom_TNF_Ins_rep")

 fit <- lmFit(eset, design)
 contrast.matrix <- makeContrasts(Mixed_rep-Mixed_Ins_rep, levels=design)