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G = C33⋊17D6  order 324 = 22·34

5th semidirect product of C33 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, A-group

Aliases: C33⋊17D6, C34⋊6C22, C32⋊6S32, C33⋊C2⋊4S3, C3⋊(C32⋊4D6), C3⋊3(S3×C3⋊S3), (C3×C3⋊S3)⋊5S3, C3⋊S3⋊2(C3⋊S3), C32⋊5(C2×C3⋊S3), (C32×C3⋊S3)⋊5C2, (C3×C33⋊C2)⋊3C2, SmallGroup(324,170)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C34 — C33⋊17D6
Chief series
C1 — C3 — C32 — C33 — C34 — C3×C33⋊C2 — C33⋊17D6
Lower central
C34 — C33⋊17D6
Upper central
C1

Generators and relations for C33⋊17D6
 G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1432 in 224 conjugacy classes, 35 normal (7 characteristic)
C1, C2, C3, C3, C22, S3, C6, C32, C32, C32, D6, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C33, C33, S32, C2×C3⋊S3, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C33⋊C2, C34, S3×C3⋊S3, C32⋊4D6, C32×C3⋊S3, C3×C33⋊C2, C33⋊17D6
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S32, C2×C3⋊S3, S3×C3⋊S3, C32⋊4D6, C33⋊17D6

Smallest permutation representation of C33⋊17D6
►On 36 points
Generators in S36
(1 23 35)(2 24 36)(3 19 31)(4 20 32)(5 21 33)(6 22 34)(7 18 28)(8 13 29)(9 14 30)(10 15 25)(11 16 26)(12 17 27)

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

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

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

33 conjugacy classes

class 1 2A2B2C3A···3F3G···3W6A6B6C6D6E6F
order12223···33···3666666
size1927272···24···4181818185454

33 irreducible representations

dim11122244
type+++++++
imageC1C2C2S3S3D6S32C32⋊4D6
kernelC33⋊17D6C32×C3⋊S3C3×C33⋊C2C3×C3⋊S3C33⋊C2C33C32C3
# reps11242698

Matrix representation of C33⋊17D6 ►in GL8(ℤ)

10000000
01000000
00100000
00010000
00000-100
00001-100
0000000-1
0000001-1
,
0-1000000
1-1000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
000-10000
001-10000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00001-100
00001000
0000000-1
0000001-1
,
-10000000
0-1000000
000-10000
00-100000
00001000
00001-100
0000001-1
0000000-1

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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[-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,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1] >;
 

C33⋊17D6 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_{17}D_6
 
% in TeX
 
G:=Group("C3^3:17D6");
 
// GroupNames label
 
G:=SmallGroup(324,170);
 
// by ID
 
G=gap.SmallGroup(324,170);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,80,579,297,1090,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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