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G = C32⋊C6⋊C8  order 432 = 24·33

The semidirect product of C32⋊C6 and C8 acting via C8/C4=C2

non-abelian, supersoluble, monomial

Aliases: C32⋊C6⋊C8, C12.87S32, He3⋊3(C2×C8), C32⋊1(S3×C8), He3⋊3C8⋊5C2, He3⋊4C8⋊4C2, C32⋊4C8⋊4S3, (C3×C12).35D6, C32⋊C12.2C4, C6.11(S3×Dic3), C3⋊Dic3.1Dic3, C4.13(C32⋊D6), (C4×He3).27C22, C3⋊S3⋊(C3⋊C8), C32⋊(C2×C3⋊C8), C3.2(S3×C3⋊C8), (C4×C3⋊S3).2S3, (C3×C6).1(C4×S3), (C4×C32⋊C6).4C2, (C2×C32⋊C6).1C4, (C2×He3).8(C2×C4), (C2×C3⋊S3).1Dic3, (C3×C6).1(C2×Dic3), C2.1(C6.S32), SmallGroup(432,76)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — C32⋊C6⋊C8
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — C4×C32⋊C6 — C32⋊C6⋊C8
Lower central
He3 — C32⋊C6⋊C8
Upper central
C1 — C4

Generators and relations for C32⋊C6⋊C8
 G = < a,b,c,d | a3=b3=c6=d8=1, ab=ba, cac-1=a-1b-1, dad-1=ab-1, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 371 in 85 conjugacy classes, 29 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C3⋊C8, C32⋊C6, C2×He3, C3×C3⋊C8, C32⋊4C8, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, S3×C3⋊C8, C12.29D6, He3⋊3C8, He3⋊4C8, C4×C32⋊C6, C32⋊C6⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C4×S3, C2×Dic3, S32, S3×C8, C2×C3⋊C8, S3×Dic3, C32⋊D6, S3×C3⋊C8, C6.S32, C32⋊C6⋊C8

Smallest permutation representation of C32⋊C6⋊C8
►On 72 points
Generators in S72
(1 31 52)(2 53 32)(3 25 54)(4 55 26)(5 27 56)(6 49 28)(7 29 50)(8 51 30)(10 64 69)(12 58 71)(14 60 65)(16 62 67)(17 47 39)(19 41 33)(21 43 35)(23 45 37)

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

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

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

44 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F8A···8H12A12B12C12D12E12F12G12H12I12J24A···24H
order1222333344446666668···81212121212121212121224···24
size11992661211992661218189···92266661212181818···18

44 irreducible representations

dim111111122222222444666
type++++++-+-+-+
imageC1C2C2C2C4C4C8S3S3Dic3D6Dic3C3⋊C8C4×S3S3×C8S32S3×Dic3S3×C3⋊C8C32⋊D6C6.S32C32⋊C6⋊C8
kernelC32⋊C6⋊C8He3⋊3C8He3⋊4C8C4×C32⋊C6C32⋊C12C2×C32⋊C6C32⋊C6C32⋊4C8C4×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C3×C6C32C12C6C3C4C2C1
# reps111122811121424112224

Matrix representation of C32⋊C6⋊C8 ►in GL10(𝔽73)

0100000000
727200000000
0001000000
007272000000
000072720000
0000100000
0000000100
000000727200
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000010000
000072720000
0000000100
000000727200
0000000001
000000007272
,
00720000000
0011000000
1010000000
72727272000000
0000001000
000000727200
0000000010
000000007272
0000100000
000072720000
,
530190000000
053019000000
390200000000
039020000000
00004600000
000027270000
00000000460
000000002727
00000046000
000000272700

G:=sub<GL(10,GF(73))| [0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,1,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0],[53,0,39,0,0,0,0,0,0,0,0,53,0,39,0,0,0,0,0,0,19,0,20,0,0,0,0,0,0,0,0,19,0,20,0,0,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,27,0,0,0,0,0,0,0,0,0,27,0,0] >;
 

C32⋊C6⋊C8 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_6\rtimes C_8
 
% in TeX
 
G:=Group("C3^2:C6:C8");
 
// GroupNames label
 
G:=SmallGroup(432,76);
 
// by ID
 
G=gap.SmallGroup(432,76);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^6=d^8=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,d*a*d^-1=a*b^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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