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## G = C4×D12order 96 = 25·3

### Direct product of C4 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — C4×D12
 Lower central C3 — C6 — C4×D12
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×D12
G = < a,b,c | a4=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 218 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], Dic3 [×2], C12 [×4], C12, D6 [×4], D6 [×4], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C4×D4, C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C4×D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], D12 [×2], C22×S3, C4×D4, S3×C2×C4, C2×D12, C4○D12, C4×D12

Smallest permutation representation of C4×D12
On 48 points
Generators in S48
(1 25 17 47)(2 26 18 48)(3 27 19 37)(4 28 20 38)(5 29 21 39)(6 30 22 40)(7 31 23 41)(8 32 24 42)(9 33 13 43)(10 34 14 44)(11 35 15 45)(12 36 16 46)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 27)(28 36)(29 35)(30 34)(31 33)(37 47)(38 46)(39 45)(40 44)(41 43)

G:=sub<Sym(48)| (1,25,17,47)(2,26,18,48)(3,27,19,37)(4,28,20,38)(5,29,21,39)(6,30,22,40)(7,31,23,41)(8,32,24,42)(9,33,13,43)(10,34,14,44)(11,35,15,45)(12,36,16,46), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,47)(38,46)(39,45)(40,44)(41,43)>;

G:=Group( (1,25,17,47)(2,26,18,48)(3,27,19,37)(4,28,20,38)(5,29,21,39)(6,30,22,40)(7,31,23,41)(8,32,24,42)(9,33,13,43)(10,34,14,44)(11,35,15,45)(12,36,16,46), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,47)(38,46)(39,45)(40,44)(41,43) );

G=PermutationGroup([(1,25,17,47),(2,26,18,48),(3,27,19,37),(4,28,20,38),(5,29,21,39),(6,30,22,40),(7,31,23,41),(8,32,24,42),(9,33,13,43),(10,34,14,44),(11,35,15,45),(12,36,16,46)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,27),(28,36),(29,35),(30,34),(31,33),(37,47),(38,46),(39,45),(40,44),(41,43)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 12A ··· 12L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 12 ··· 12 size 1 1 1 1 6 6 6 6 2 1 1 1 1 2 2 2 2 6 6 6 6 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 C4×S3 D12 C4○D12 kernel C4×D12 C4⋊Dic3 D6⋊C4 C4×C12 S3×C2×C4 C2×D12 D12 C42 C12 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 2 3 2 4 4 4

Matrix representation of C4×D12 in GL3(𝔽13) generated by

 8 0 0 0 8 0 0 0 8
,
 12 0 0 0 3 10 0 3 6
,
 12 0 0 0 1 1 0 0 12
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[12,0,0,0,3,3,0,10,6],[12,0,0,0,1,0,0,1,12] >;

C4×D12 in GAP, Magma, Sage, TeX

C_4\times D_{12}
% in TeX

G:=Group("C4xD12");
// GroupNames label

G:=SmallGroup(96,80);
// by ID

G=gap.SmallGroup(96,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,50,2309]);
// Polycyclic

G:=Group<a,b,c|a^4=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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