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## G = C2×C6.D12order 288 = 25·32

### Direct product of C2 and C6.D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C6.D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C6.D12 — C2×C6.D12
 Lower central C32 — C3×C6 — C2×C6.D12
 Upper central C1 — C23

Generators and relations for C2×C6.D12
G = < a,b,c,d | a2=b6=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 1474 in 331 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], C22 [×16], S3 [×16], C6 [×14], C6 [×7], C2×C4 [×8], C23, C23 [×10], C32, Dic3 [×4], C12 [×4], D6 [×60], C2×C6 [×14], C2×C6 [×7], C22⋊C4 [×4], C22×C4 [×2], C24, C3⋊S3 [×4], C3×C6, C3×C6 [×6], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×8], C22×S3 [×34], C22×C6 [×2], C22×C6, C2×C22⋊C4, C3×Dic3 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62, C62 [×6], D6⋊C4 [×8], C22×Dic3 [×2], C22×C12 [×2], S3×C23 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C22×C3⋊S3 [×6], C22×C3⋊S3 [×4], C2×C62, C2×D6⋊C4 [×2], C6.D12 [×4], Dic3×C2×C6 [×2], C23×C3⋊S3, C2×C6.D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C22×S3 [×2], C2×C22⋊C4, S32, D6⋊C4 [×8], S3×C2×C4 [×2], C2×D12 [×2], C2×C3⋊D4 [×2], C6.D6 [×2], C3⋊D12 [×4], C2×S32, C2×D6⋊C4 [×2], C6.D12 [×4], C2×C6.D6, C2×C3⋊D12 [×2], C2×C6.D12

Smallest permutation representation of C2×C6.D12
On 48 points
Generators in S48
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(1 36 9 32 5 28)(2 29 6 33 10 25)(3 26 11 34 7 30)(4 31 8 35 12 27)(13 43 21 39 17 47)(14 48 18 40 22 44)(15 45 23 41 19 37)(16 38 20 42 24 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 45)(2 18)(3 43)(4 16)(5 41)(6 14)(7 39)(8 24)(9 37)(10 22)(11 47)(12 20)(13 26)(15 36)(17 34)(19 32)(21 30)(23 28)(25 40)(27 38)(29 48)(31 46)(33 44)(35 42)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 3C 4A ··· 4H 6A ··· 6N 6O ··· 6U 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 18 18 18 18 2 2 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 S32 C6.D6 C3⋊D12 C2×S32 kernel C2×C6.D12 C6.D12 Dic3×C2×C6 C23×C3⋊S3 C22×C3⋊S3 C22×Dic3 C62 C2×Dic3 C22×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 # reps 1 4 2 1 8 2 4 4 2 8 8 8 1 2 4 1

Matrix representation of C2×C6.D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 8 5
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,5],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;

C2×C6.D12 in GAP, Magma, Sage, TeX

C_2\times C_6.D_{12}
% in TeX

G:=Group("C2xC6.D12");
// GroupNames label

G:=SmallGroup(288,611);
// by ID

G=gap.SmallGroup(288,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,176,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
// generators/relations

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