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G = C33⋊7D4  order 216 = 23·33

4th semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C33⋊7D4, C32⋊10D12, C6.13S32, (S3×C6)⋊4S3, D6⋊2(C3⋊S3), C3⋊Dic3⋊4S3, (C3×C6).32D6, C3⋊2(C3⋊D12), C32⋊7(C3⋊D4), C3⋊1(C32⋊7D4), (C32×C6).10C22, (S3×C3×C6)⋊4C2, C6.5(C2×C3⋊S3), C2.5(S3×C3⋊S3), (C3×C3⋊Dic3)⋊2C2, (C2×C33⋊C2)⋊1C2, SmallGroup(216,128)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊7D4
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊7D4
Lower central
C33 — C32×C6 — C33⋊7D4
Upper central
C1 — C2

Generators and relations for C33⋊7D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 812 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, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C3⋊D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, S3×C32, C33⋊C2, C32×C6, C3⋊D12, C32⋊7D4, C3×C3⋊Dic3, S3×C3×C6, C2×C33⋊C2, C33⋊7D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, C3⋊D12, C32⋊7D4, S3×C3⋊S3, C33⋊7D4

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

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

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

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

C33⋊7D4 is a maximal subgroup of
 S3×C3⋊D12  D6⋊4S32  (S3×C6).D6  D6.3S32  D12⋊(C3⋊S3)  C12.73S32  C12.58S32  C3⋊S3×D12  C62.90D6  S3×C32⋊7D4  C62⋊23D6
C33⋊7D4 is a maximal quotient of
 C33⋊7D8  C33⋊14SD16  C33⋊15SD16  C33⋊7Q16  C62.77D6  C62.79D6  C62.82D6

33 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J···6Q12A12B
order12223···3333346···666666···61212
size116542···24444182···244446···61818

33 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2S3S3D4D6D12C3⋊D4S32C3⋊D12
kernelC33⋊7D4C3×C3⋊Dic3S3×C3×C6C2×C33⋊C2C3⋊Dic3S3×C6C33C3×C6C32C32C6C3
# reps111114152844

Matrix representation of C33⋊7D4 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000011000
000011100
0000001212
00000010
,
10000000
01000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
10000000
01000000
000120000
001120000
00001000
00000100
00000010
00000001
,
73000000
56000000
001200000
000120000
000011000
000001200
00000010
0000001212
,
120000000
91000000
00010000
00100000
000012300
00000100
00000010
0000001212

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[7,5,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[12,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;
 

C33⋊7D4 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_7D_4
 
% in TeX
 
G:=Group("C3^3:7D4");
 
// GroupNames label
 
G:=SmallGroup(216,128);
 
// by ID
 
G=gap.SmallGroup(216,128);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,201,730,5189]);
 
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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