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G = C2×C12⋊S3order 144 = 24·32

Direct product of C2 and C12⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12⋊S3, C61D12, C126D6, C62.33C22, (C3×C6)⋊5D4, (C2×C12)⋊3S3, (C6×C12)⋊4C2, C32(C2×D12), C329(C2×D4), (C2×C6).38D6, (C3×C12)⋊6C22, C6.32(C22×S3), (C3×C6).31C23, C42(C2×C3⋊S3), (C2×C4)⋊2(C3⋊S3), (C22×C3⋊S3)⋊3C2, (C2×C3⋊S3)⋊5C22, C2.4(C22×C3⋊S3), C22.10(C2×C3⋊S3), SmallGroup(144,170)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C12⋊S3
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3 — C2×C12⋊S3
C32C3×C6 — C2×C12⋊S3
C1C22C2×C4

Generators and relations for C2×C12⋊S3
 G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 642 in 162 conjugacy classes, 59 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×16], C6 [×12], C2×C4, D4 [×4], C23 [×2], C32, C12 [×8], D6 [×32], C2×C6 [×4], C2×D4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], D12 [×16], C2×C12 [×4], C22×S3 [×8], C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C2×D12 [×4], C12⋊S3 [×4], C6×C12, C22×C3⋊S3 [×2], C2×C12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C2×C12⋊S3

Smallest permutation representation of C2×C12⋊S3
On 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 71)(14 72)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 49)(36 50)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 29 17)(2 30 18)(3 31 19)(4 32 20)(5 33 21)(6 34 22)(7 35 23)(8 36 24)(9 25 13)(10 26 14)(11 27 15)(12 28 16)(37 52 72)(38 53 61)(39 54 62)(40 55 63)(41 56 64)(42 57 65)(43 58 66)(44 59 67)(45 60 68)(46 49 69)(47 50 70)(48 51 71)
(1 40)(2 39)(3 38)(4 37)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 60)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 72)(33 71)(34 70)(35 69)(36 68)

G:=sub<Sym(72)| (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,37)(11,38)(12,39)(13,71)(14,72)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,49)(36,50), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,17)(2,30,18)(3,31,19)(4,32,20)(5,33,21)(6,34,22)(7,35,23)(8,36,24)(9,25,13)(10,26,14)(11,27,15)(12,28,16)(37,52,72)(38,53,61)(39,54,62)(40,55,63)(41,56,64)(42,57,65)(43,58,66)(44,59,67)(45,60,68)(46,49,69)(47,50,70)(48,51,71), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,60)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,72)(33,71)(34,70)(35,69)(36,68)>;

G:=Group( (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,37)(11,38)(12,39)(13,71)(14,72)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,49)(36,50), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,17)(2,30,18)(3,31,19)(4,32,20)(5,33,21)(6,34,22)(7,35,23)(8,36,24)(9,25,13)(10,26,14)(11,27,15)(12,28,16)(37,52,72)(38,53,61)(39,54,62)(40,55,63)(41,56,64)(42,57,65)(43,58,66)(44,59,67)(45,60,68)(46,49,69)(47,50,70)(48,51,71), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,60)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,72)(33,71)(34,70)(35,69)(36,68) );

G=PermutationGroup([(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,37),(11,38),(12,39),(13,71),(14,72),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,49),(36,50)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,29,17),(2,30,18),(3,31,19),(4,32,20),(5,33,21),(6,34,22),(7,35,23),(8,36,24),(9,25,13),(10,26,14),(11,27,15),(12,28,16),(37,52,72),(38,53,61),(39,54,62),(40,55,63),(41,56,64),(42,57,65),(43,58,66),(44,59,67),(45,60,68),(46,49,69),(47,50,70),(48,51,71)], [(1,40),(2,39),(3,38),(4,37),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,60),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,72),(33,71),(34,70),(35,69),(36,68)])

C2×C12⋊S3 is a maximal subgroup of
C12.70D12  C6.17D24  C62.113D4  C62.84D4  C12.19D12  (C6×C12)⋊C4  D1218D6  C12.28D12  Dic35D12  C62.67C23  C127D12  Dic33D12  C12⋊D12  D65D12  C124D12  C1226C2  C6212D4  C62.228C23  C62.237C23  C62.238C23  C123D12  C243D6  C6219D4  C62.258C23  C62.262C23  C62.73D4  C2×S3×D12  D1227D6  C2×D4×C3⋊S3  C62.154C23
C2×C12⋊S3 is a maximal quotient of
C126Dic6  C124D12  C1226C2  C6212D4  C62.69D4  C123D12  C12.31D12  C24.78D6  C243D6  C24.5D6  C6219D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A···6L12A···12P
order122222223333446···612···12
size1111181818182222222···22···2

42 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2S3D4D6D6D12
kernelC2×C12⋊S3C12⋊S3C6×C12C22×C3⋊S3C2×C12C3×C6C12C2×C6C6
# reps1412428416

Matrix representation of C2×C12⋊S3 in GL5(𝔽13)

120000
01000
00100
00010
00001
,
120000
06300
010300
00063
000103
,
10000
00100
0121200
00010
00001
,
10000
01000
0121200
000012
000120

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,6,10,0,0,0,3,3,0,0,0,0,0,6,10,0,0,0,3,3],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

C2×C12⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC12:S3");
// GroupNames label

G:=SmallGroup(144,170);
// by ID

G=gap.SmallGroup(144,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,50,964,3461]);
// Polycyclic

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

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