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G = C33⋊8D4  order 216 = 23·33

5th semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C33⋊8D4, C32⋊5D12, C6.14S32, Dic3⋊(C3⋊S3), (C3×C6).33D6, (C3×Dic3)⋊1S3, C3⋊2(C12⋊S3), C3⋊1(C3⋊D12), C32⋊10(C3⋊D4), (C32×Dic3)⋊1C2, (C32×C6).11C22, (C6×C3⋊S3)⋊3C2, (C2×C3⋊S3)⋊5S3, C6.6(C2×C3⋊S3), C2.6(S3×C3⋊S3), (C2×C33⋊C2)⋊2C2, SmallGroup(216,129)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊8D4
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8D4
Lower central
C33 — C32×C6 — C33⋊8D4
Upper central
C1 — C2

Generators and relations for C33⋊8D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 852 in 136 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C3⋊D4, C33, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C3⋊D12, C12⋊S3, C32×Dic3, C6×C3⋊S3, C2×C33⋊C2, C33⋊8D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, C3⋊D12, C12⋊S3, S3×C3⋊S3, C33⋊8D4

Smallest permutation representation of C33⋊8D4
►On 36 points
Generators in S36
(1 21 8)(2 22 5)(3 23 6)(4 24 7)(9 18 32)(10 19 29)(11 20 30)(12 17 31)(13 36 26)(14 33 27)(15 34 28)(16 35 25)

(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 26 31)(6 27 32)(7 28 29)(8 25 30)(17 22 36)(18 23 33)(19 24 34)(20 21 35)

(1 8 21)(2 22 5)(3 6 23)(4 24 7)(9 32 18)(10 19 29)(11 30 20)(12 17 31)(13 36 26)(14 27 33)(15 34 28)(16 25 35)

(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)

(2 4)(5 24)(6 23)(7 22)(8 21)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(29 36)(30 35)(31 34)(32 33)
 
G:=sub<Sym(36)| (1,21,8)(2,22,5)(3,23,6)(4,24,7)(9,18,32)(10,19,29)(11,20,30)(12,17,31)(13,36,26)(14,33,27)(15,34,28)(16,35,25), (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,26,31)(6,27,32)(7,28,29)(8,25,30)(17,22,36)(18,23,33)(19,24,34)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (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), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33)>;
 
G:=Group( (1,21,8)(2,22,5)(3,23,6)(4,24,7)(9,18,32)(10,19,29)(11,20,30)(12,17,31)(13,36,26)(14,33,27)(15,34,28)(16,35,25), (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,26,31)(6,27,32)(7,28,29)(8,25,30)(17,22,36)(18,23,33)(19,24,34)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (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), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33) );
 
G=PermutationGroup([[(1,21,8),(2,22,5),(3,23,6),(4,24,7),(9,18,32),(10,19,29),(11,20,30),(12,17,31),(13,36,26),(14,33,27),(15,34,28),(16,35,25)], [(1,16,11),(2,13,12),(3,14,9),(4,15,10),(5,26,31),(6,27,32),(7,28,29),(8,25,30),(17,22,36),(18,23,33),(19,24,34),(20,21,35)], [(1,8,21),(2,22,5),(3,6,23),(4,24,7),(9,32,18),(10,19,29),(11,30,20),(12,17,31),(13,36,26),(14,27,33),(15,34,28),(16,25,35)], [(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)], [(2,4),(5,24),(6,23),(7,22),(8,21),(9,14),(10,13),(11,16),(12,15),(17,28),(18,27),(19,26),(20,25),(29,36),(30,35),(31,34),(32,33)]])
 

C33⋊8D4 is a maximal subgroup of
 S3×C3⋊D12  C3⋊S3⋊4D12  D6.3S32  Dic3.S32  C12.39S32  C12.40S32  C12.73S32  S3×C12⋊S3  C62.93D6  C3⋊S3×C3⋊D4  C62⋊23D6
C33⋊8D4 is a maximal quotient of
 C33⋊8D8  C33⋊16SD16  C33⋊17SD16  C33⋊8Q16  C62.78D6  C62.79D6  C62.80D6

33 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J6K12A···12H
order12223···3333346···666666612···12
size1118542···2444462···2444418186···6

33 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2S3S3D4D6D12C3⋊D4S32C3⋊D12
kernelC33⋊8D4C32×Dic3C6×C3⋊S3C2×C33⋊C2C3×Dic3C2×C3⋊S3C33C3×C6C32C32C6C3
# reps111141158244

Matrix representation of C33⋊8D4 ►in GL8(ℤ)

10000000
01000000
00100000
00010000
00000-100
00001-100
00000010
00000001
,
10000000
01000000
00-1-10000
00100000
00000-100
00001-100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000000-1
0000001-1
,
0-1000000
10000000
00-100000
000-10000
0000-1000
00000-100
00000001
00000010
,
10000000
0-1000000
00100000
00-1-10000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C33⋊8D4 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_8D_4
 
% in TeX
 
G:=Group("C3^3:8D4");
 
// GroupNames label
 
G:=SmallGroup(216,129);
 
// by ID
 
G=gap.SmallGroup(216,129);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,201,730,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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