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G = C2×C4×C3⋊S3order 144 = 24·32

Direct product of C2×C4 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C4×C3⋊S3, C127D6, C62.32C22, C62(C4×S3), (C2×C12)⋊5S3, (C6×C12)⋊9C2, (C2×C6).37D6, (C3×C12)⋊9C22, C326(C22×C4), C6.31(C22×S3), (C3×C6).30C23, C3⋊Dic38C22, C33(S3×C2×C4), (C3×C6)⋊5(C2×C4), (C2×C3⋊Dic3)⋊9C2, C22.9(C2×C3⋊S3), C2.1(C22×C3⋊S3), (C22×C3⋊S3).6C2, (C2×C3⋊S3).21C22, SmallGroup(144,169)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C4×C3⋊S3
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3 — C2×C4×C3⋊S3
C32 — C2×C4×C3⋊S3
C1C2×C4

Generators and relations for C2×C4×C3⋊S3
 G = < a,b,c,d,e | a2=b4=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 466 in 162 conjugacy classes, 67 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C4×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, C2×C4×C3⋊S3

Smallest permutation representation of C2×C4×C3⋊S3
On 72 points
Generators in S72
(1 18)(2 19)(3 20)(4 17)(5 68)(6 65)(7 66)(8 67)(9 44)(10 41)(11 42)(12 43)(13 62)(14 63)(15 64)(16 61)(21 47)(22 48)(23 45)(24 46)(25 60)(26 57)(27 58)(28 59)(29 55)(30 56)(31 53)(32 54)(33 50)(34 51)(35 52)(36 49)(37 70)(38 71)(39 72)(40 69)
(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 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 50 69)(10 51 70)(11 52 71)(12 49 72)(17 46 54)(18 47 55)(19 48 56)(20 45 53)(25 66 63)(26 67 64)(27 68 61)(28 65 62)(33 40 44)(34 37 41)(35 38 42)(36 39 43)
(1 33 16)(2 34 13)(3 35 14)(4 36 15)(5 29 44)(6 30 41)(7 31 42)(8 32 43)(9 68 55)(10 65 56)(11 66 53)(12 67 54)(17 49 64)(18 50 61)(19 51 62)(20 52 63)(21 40 58)(22 37 59)(23 38 60)(24 39 57)(25 45 71)(26 46 72)(27 47 69)(28 48 70)
(1 18)(2 19)(3 20)(4 17)(5 69)(6 70)(7 71)(8 72)(9 58)(10 59)(11 60)(12 57)(13 51)(14 52)(15 49)(16 50)(21 55)(22 56)(23 53)(24 54)(25 42)(26 43)(27 44)(28 41)(29 47)(30 48)(31 45)(32 46)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)

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

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

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

C2×C4×C3⋊S3 is a maximal subgroup of
C12.78D12  C12.60D12  C62.6(C2×C4)  (C6×C12)⋊2C4  C3⋊C820D6  C62.19C23  C62.23C23  C62.35C23  C12.30D12  C62.44C23  C62.51C23  C62.53C23  C62.70C23  C62.82C23  C122D12  C62.91C23  C12216C2  C62.225C23  C62.227C23  C62.228C23  C62.236C23  C62.237C23  C62.238C23  C123D12  C62.240C23  C12.31D12  C62.256C23  C62.261C23  C3⋊S3⋊M4(2)  (C6×C12)⋊5C4  S32×C2×C4  D1223D6
C2×C4×C3⋊S3 is a maximal quotient of
C12216C2  C62.221C23  C62.225C23  C62.231C23  C62.236C23  C62.237C23  C24.95D6  C24.47D6

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B4C4D4E4F4G4H6A···6L12A···12P
order122222223333444444446···612···12
size111199992222111199992···22···2

48 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4S3D6D6C4×S3
kernelC2×C4×C3⋊S3C4×C3⋊S3C2×C3⋊Dic3C6×C12C22×C3⋊S3C2×C3⋊S3C2×C12C12C2×C6C6
# reps14111848416

Matrix representation of C2×C4×C3⋊S3 in GL4(𝔽13) generated by

12000
01200
0010
0001
,
5000
0500
0050
0005
,
0100
121200
0010
0001
,
1000
0100
001212
0010
,
121200
0100
0010
001212
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[12,0,0,0,12,1,0,0,0,0,1,12,0,0,0,12] >;

C2×C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2xC4xC3:S3");
// GroupNames label

G:=SmallGroup(144,169);
// by ID

G=gap.SmallGroup(144,169);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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