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## G = C2×D12⋊S3order 288 = 25·32

### Direct product of C2 and D12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×D12⋊S3
 Lower central C32 — C3×C6 — C2×D12⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D12⋊S3
G = < a,b,c,d,e | a2=b12=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 1234 in 355 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×12], C6 [×2], C6 [×4], C6 [×7], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×6], D6 [×4], D6 [×18], C2×C6 [×2], C2×C6 [×9], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×4], C4×S3 [×20], D12 [×4], D12 [×8], C2×Dic3 [×2], C2×Dic3 [×11], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C3×Q8 [×4], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C2×C4○D4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×Dic6, S3×C2×C4 [×5], C2×D12, C2×D12 [×2], D42S3 [×8], Q83S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, S3×Dic3 [×8], C3⋊D12 [×8], C3×Dic6 [×4], C3×D12 [×4], C6×Dic3 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C2×D42S3, C2×Q83S3, D12⋊S3 [×8], C2×S3×Dic3 [×2], C2×C3⋊D12 [×2], C6×Dic6, C6×D12, C2×C4×C3⋊S3, C2×D12⋊S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, D42S3 [×2], Q83S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×D42S3, C2×Q83S3, D12⋊S3 [×2], C22×S32, C2×D12⋊S3

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 6 6 18 18 2 2 4 2 2 6 6 6 6 9 9 9 9 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 S32 D4⋊2S3 Q8⋊3S3 C2×S32 C2×S32 D12⋊S3 kernel C2×D12⋊S3 D12⋊S3 C2×S3×Dic3 C2×C3⋊D12 C6×Dic6 C6×D12 C2×C4×C3⋊S3 C2×Dic6 C2×D12 Dic6 D12 C2×Dic3 C2×C12 C22×S3 C3×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 1 4 4 2 2 2 4 1 2 2 2 1 4

Matrix representation of C2×D12⋊S3 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 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 1 0 0 0 0 0 0 8 0 0 0 0 0 5 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 0 1 0 0 0 0 0 0 12 11 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 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 1 1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;

C2×D12⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD12:S3");
// GroupNames label

G:=SmallGroup(288,944);
// by ID

G=gap.SmallGroup(288,944);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^3=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,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations

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