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G = C8×C32⋊C6  order 432 = 24·33

Direct product of C8 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C8×C32⋊C6, C3⋊S3⋊2C24, (C3×C24)⋊4C6, (C3×C24)⋊5S3, He3⋊6(C2×C8), C6.7(S3×C12), C3.2(S3×C24), (C8×He3)⋊6C2, C32⋊4(S3×C8), C24.19(C3×S3), C12.86(S3×C6), C32⋊2(C2×C24), C32⋊4C8⋊6C6, (C3×C12).56D6, He3⋊3C8⋊13C2, C32⋊C12.4C4, C3⋊Dic3.3C12, (C4×He3).41C22, (C8×C3⋊S3)⋊C3, (C4×C3⋊S3).3C6, (C2×C3⋊S3).3C12, (C3×C6).10(C4×S3), (C3×C6).1(C2×C12), C2.1(C4×C32⋊C6), (C3×C12).13(C2×C6), (C4×C32⋊C6).6C2, (C2×C32⋊C6).4C4, C4.12(C2×C32⋊C6), (C2×He3).17(C2×C4), SmallGroup(432,115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C8×C32⋊C6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — C4×C32⋊C6 — C8×C32⋊C6
Lower central
C32 — C8×C32⋊C6
Upper central
C1 — C8

Generators and relations for C8×C32⋊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, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C32⋊C6, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×C24, C3×C24, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, S3×C24, C8×C3⋊S3, He3⋊3C8, C8×He3, C4×C32⋊C6, C8×C32⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, S3×C6, S3×C8, C2×C24, C32⋊C6, S3×C12, C2×C32⋊C6, S3×C24, C4×C32⋊C6, C8×C32⋊C6

Smallest permutation representation of C8×C32⋊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++++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24S3D6C3×S3C4×S3S3×C6S3×C8S3×C12S3×C24C32⋊C6C2×C32⋊C6C4×C32⋊C6C8×C32⋊C6
kernelC8×C32⋊C6He3⋊3C8C8×He3C4×C32⋊C6C8×C3⋊S3C32⋊C12C2×C32⋊C6C32⋊4C8C3×C24C4×C3⋊S3C32⋊C6C3⋊Dic3C2×C3⋊S3C3⋊S3C3×C24C3×C12C24C3×C6C12C32C6C3C8C4C2C1
# reps111122222284416112224481124

Matrix representation of C8×C32⋊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] >;
 

C8×C32⋊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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