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G = C8xC32:C6  order 432 = 24·33

Direct product of C8 and C32:C6

direct product, metabelian, supersoluble, monomial

Aliases: C8xC32:C6, C3:S3:2C24, (C3xC24):4C6, (C3xC24):5S3, He3:6(C2xC8), C6.7(S3xC12), C3.2(S3xC24), (C8xHe3):6C2, C32:4(S3xC8), C24.19(C3xS3), C12.86(S3xC6), C32:2(C2xC24), C32:4C8:6C6, (C3xC12).56D6, He3:3C8:13C2, C32:C12.4C4, C3:Dic3.3C12, (C4xHe3).41C22, (C8xC3:S3):C3, (C4xC3:S3).3C6, (C2xC3:S3).3C12, (C3xC6).10(C4xS3), (C3xC6).1(C2xC12), C2.1(C4xC32:C6), (C3xC12).13(C2xC6), (C4xC32:C6).6C2, (C2xC32:C6).4C4, C4.12(C2xC32:C6), (C2xHe3).17(C2xC4), SmallGroup(432,115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C8xC32:C6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C4xHe3 — C4xC32:C6 — C8xC32:C6
Lower central
C32 — C8xC32:C6
Upper central
C1 — C8

Generators and relations for C8xC32:C6
 G = < a,b,c,d | a8=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 321 in 85 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C24, C24, C4xS3, C2xC12, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, S3xC8, C2xC24, C32:C6, C2xHe3, C3xC3:C8, C32:4C8, C3xC24, C3xC24, S3xC12, C4xC3:S3, C32:C12, C4xHe3, C2xC32:C6, S3xC24, C8xC3:S3, He3:3C8, C8xHe3, C4xC32:C6, C8xC32:C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2xC4, C12, D6, C2xC6, C2xC8, C3xS3, C24, C4xS3, C2xC12, S3xC6, S3xC8, C2xC24, C32:C6, S3xC12, C2xC32:C6, S3xC24, C4xC32:C6, C8xC32:C6

Smallest permutation representation of C8xC32:C6
►On 72 points
Generators in S72
(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 68 56)(2 69 49)(3 70 50)(4 71 51)(5 72 52)(6 65 53)(7 66 54)(8 67 55)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 41 27)(18 42 28)(19 43 29)(20 44 30)(21 45 31)(22 46 32)(23 47 25)(24 48 26)

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

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

80 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B6C6D6E6F6G6H6I6J8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G···12L12M12N12O12P24A24B24C24D24E···24L24M···24X24Y···24AF
order1222333333444466666666668888888812121212121212···12121212122424242424···2424···2424···24
size119923366611992336669999111199992233336···6999922223···36···69···9

80 irreducible representations

dim11111111111111222222226666
type++++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24S3D6C3xS3C4xS3S3xC6S3xC8S3xC12S3xC24C32:C6C2xC32:C6C4xC32:C6C8xC32:C6
kernelC8xC32:C6He3:3C8C8xHe3C4xC32:C6C8xC3:S3C32:C12C2xC32:C6C32:4C8C3xC24C4xC3:S3C32:C6C3:Dic3C2xC3:S3C3:S3C3xC24C3xC12C24C3xC6C12C32C6C3C8C4C2C1
# reps111122222284416112224481124

Matrix representation of C8xC32:C6 ►in GL8(F73)

220000000
022000000
004600000
000460000
000046000
000004600
000000460
000000046
,
721000000
720000000
00001000
00000100
00000010
00000001
00100000
00010000
,
10000000
01000000
000720000
001720000
000007200
000017200
000000072
000000172
,
09000000
90000000
00010000
00100000
000000720
000000721
000017200
000007200

G:=sub<GL(8,GF(73))| [22,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0] >;
 

C8xC32:C6 in GAP, Magma, Sage, TeX 

C_8\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C8xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(432,115);
 
// by ID
 
G=gap.SmallGroup(432,115);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,4037,2035,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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