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## G = C2×S3×D12order 288 = 25·32

### Direct product of C2, S3 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×S3×D12
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — C2×S3×D12
 Lower central C32 — C3×C6 — C2×S3×D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×D12
G = < a,b,c,d,e | a2=b3=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2242 in 499 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×38], S3 [×4], S3 [×16], C6 [×2], C6 [×4], C6 [×11], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×10], D6 [×60], C2×C6 [×2], C2×C6 [×15], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3×S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], D12 [×4], D12 [×20], C2×Dic3, C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C3×D4 [×4], C22×S3, C22×S3 [×2], C22×S3 [×38], C22×C6 [×3], C22×D4, C3×Dic3 [×2], C3×C12 [×2], S32 [×16], S3×C6 [×10], S3×C6 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, S3×C2×C4, C2×D12, C2×D12 [×13], S3×D4 [×8], C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23 [×4], C3⋊D12 [×8], S3×C12 [×4], C3×D12 [×4], C6×Dic3, C12⋊S3 [×4], C6×C12, C2×S32 [×8], C2×S32 [×8], S3×C2×C6, S3×C2×C6 [×2], C22×C3⋊S3 [×2], C22×D12, C2×S3×D4, S3×D12 [×8], C2×C3⋊D12 [×2], S3×C2×C12, C6×D12, C2×C12⋊S3, C22×S32 [×2], C2×S3×D12
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, D12 [×4], C22×S3 [×14], C22×D4, S32, C2×D12 [×6], S3×D4 [×2], S3×C23 [×2], C2×S32 [×3], C22×D12, C2×S3×D4, S3×D12 [×2], C22×S32, C2×S3×D12

Smallest permutation representation of C2×S3×D12
On 48 points
Generators in S48
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)
(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 13)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 48)(36 47)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 1 1 3 3 3 3 6 6 6 6 18 18 18 18 2 2 4 2 2 6 6 2 ··· 2 4 4 4 6 6 6 6 12 12 12 12 2 2 2 2 4 ··· 4 6 6 6 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D6 D12 S32 S3×D4 C2×S32 C2×S32 S3×D12 kernel C2×S3×D12 S3×D12 C2×C3⋊D12 S3×C2×C12 C6×D12 C2×C12⋊S3 C22×S32 S3×C2×C4 C2×D12 S3×C6 C4×S3 D12 C2×Dic3 C2×C12 C22×S3 D6 C2×C4 C6 C4 C22 C2 # reps 1 8 2 1 1 1 2 1 1 4 4 4 1 2 3 8 1 2 2 1 4

Matrix representation of C2×S3×D12 in GL6(ℤ)

 1 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 0 0 0 0 0 -1
,
 0 1 0 0 0 0 -1 -1 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
,
 1 0 0 0 0 0 -1 -1 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
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 2 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×S3×D12 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{12}
% in TeX

G:=Group("C2xS3xD12");
// GroupNames label

G:=SmallGroup(288,951);
// by ID

G=gap.SmallGroup(288,951);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,346,80,1356,9414]);
// Polycyclic

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

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