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G = C2xC4xC32:C6  order 432 = 24·33

Direct product of C2xC4 and C32:C6

direct product, metabelian, supersoluble, monomial

Aliases: C2xC4xC32:C6, C62.34D6, (C6xC12):7S3, (C6xC12):5C6, (C3xC12):8D6, C6.21(S3xC12), C12.97(S3xC6), He3:5(C22xC4), C62.10(C2xC6), (C4xHe3):8C22, C32:1(C22xC12), C32:C12:10C22, (C2xHe3).20C23, (C22xHe3).27C22, C3:S3:(C2xC12), (C4xC3:S3):5C6, (C3xC6):3(C4xS3), C6.24(S3xC2xC6), C3.2(S3xC2xC12), (C2xC4xHe3):8C2, C32:4(S3xC2xC4), (C2xC3:S3):2C12, (C3xC12):3(C2xC6), (C3xC6):1(C2xC12), (C2xC6).54(S3xC6), (C2xHe3):4(C2xC4), C3:Dic3:4(C2xC6), (C2xC3:Dic3):5C6, (C2xC12).36(C3xS3), (C22xC3:S3).3C6, (C3xC6).2(C22xC6), (C2xC32:C12):11C2, (C3xC6).20(C22xS3), C22.9(C2xC32:C6), C2.1(C22xC32:C6), (C22xC32:C6).4C2, (C2xC32:C6).13C22, (C2xC4xC3:S3):C3, (C2xC3:S3).11(C2xC6), SmallGroup(432,349)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2xC4xC32:C6
Chief series
C1 — C3 — C32 — C3xC6 — C2xHe3 — C2xC32:C6 — C22xC32:C6 — C2xC4xC32:C6
Lower central
C32 — C2xC4xC32:C6
Upper central
C1 — C2xC4

Generators and relations for C2xC4xC32:C6
 G = < a,b,c,d,e | a2=b4=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 817 in 205 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C23, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C22xC4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C62, C62, S3xC2xC4, C22xC12, C32:C6, C2xHe3, C2xHe3, S3xC12, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, C6xC12, S3xC2xC6, C22xC3:S3, C32:C12, C4xHe3, C2xC32:C6, C22xHe3, S3xC2xC12, C2xC4xC3:S3, C4xC32:C6, C2xC32:C12, C2xC4xHe3, C22xC32:C6, C2xC4xC32:C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, C12, D6, C2xC6, C22xC4, C3xS3, C4xS3, C2xC12, C22xS3, C22xC6, S3xC6, S3xC2xC4, C22xC12, C32:C6, S3xC12, S3xC2xC6, C2xC32:C6, S3xC2xC12, C4xC32:C6, C22xC32:C6, C2xC4xC32:C6

Smallest permutation representation of C2xC4xC32:C6
►On 72 points
Generators in S72
(1 21)(2 22)(3 14)(4 13)(5 7)(6 8)(9 20)(10 19)(11 24)(12 23)(15 17)(16 18)(25 56)(26 57)(27 58)(28 59)(29 60)(30 55)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 63)(38 64)(39 65)(40 66)(41 61)(42 62)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)

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

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F4A4B4C4D4E4F4G4H6A6B6C6D···6I6J···6R6S···6Z12A12B12C12D12E···12L12M···12X12Y···12AF
order12222222333333444444446666···66···66···61212121212···1212···1212···12
size11119999233666111199992223···36···69···922223···36···69···9

80 irreducible representations

dim111111111111222222226666
type+++++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12S3D6D6C3xS3C4xS3S3xC6S3xC6S3xC12C32:C6C2xC32:C6C2xC32:C6C4xC32:C6
kernelC2xC4xC32:C6C4xC32:C6C2xC32:C12C2xC4xHe3C22xC32:C6C2xC4xC3:S3C2xC32:C6C4xC3:S3C2xC3:Dic3C6xC12C22xC3:S3C2xC3:S3C6xC12C3xC12C62C2xC12C3xC6C12C2xC6C6C2xC4C4C22C2
# reps1411128822216121244281214

Matrix representation of C2xC4xC32:C6 ►in GL8(F13)

10000000
01000000
001200000
000120000
000012000
000001200
000000120
000000012
,
50000000
05000000
001200000
000120000
000012000
000001200
000000120
000000012
,
01000000
1212000000
00000100
0000121200
00000001
0000001212
00010000
0012120000
,
10000000
01000000
0012120000
00100000
0000121200
00001000
0000001212
00000010
,
40000000
99000000
00100000
0012120000
00000001
00000010
0000121200
00000100

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[4,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C2xC4xC32:C6 in GAP, Magma, Sage, TeX 

C_2\times C_4\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C2xC4xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(432,349);
 
// by ID
 
G=gap.SmallGroup(432,349);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,4037,1034,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^3=e^6=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^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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