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## G = C22×D12order 96 = 25·3

### Direct product of C22 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C22×D12
 Lower central C3 — C6 — C22×D12
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×D12
G = < a,b,c,d | a2=b2=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 578 in 236 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], C12 [×4], D6 [×8], D6 [×24], C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], D12 [×16], C2×C12 [×6], C22×S3 [×12], C22×S3 [×8], C22×C6, C22×D4, C2×D12 [×12], C22×C12, S3×C23 [×2], C22×D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, C22×D12

Smallest permutation representation of C22×D12
On 48 points
Generators in S48
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

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

36 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A 4B 4C 4D 6A ··· 6G 12A ··· 12H order 1 2 ··· 2 2 ··· 2 3 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 2 2 2 2 2 2 ··· 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 kernel C22×D12 C2×D12 C22×C12 S3×C23 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 2 1 4 6 1 8

Matrix representation of C22×D12 in GL5(ℤ)

 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 -1 -1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 -1 0
,
 -1 0 0 0 0 0 1 1 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,1,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

C22×D12 in GAP, Magma, Sage, TeX

C_2^2\times D_{12}
% in TeX

G:=Group("C2^2xD12");
// GroupNames label

G:=SmallGroup(96,207);
// by ID

G=gap.SmallGroup(96,207);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,579,69,2309]);
// Polycyclic

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

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