Copied to
clipboard

G = C3317D6order 324 = 22·34

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

metabelian, supersoluble, monomial, A-group

Aliases: C3317D6, C346C22, C326S32, C33⋊C24S3, C3⋊(C324D6), C33(S3×C3⋊S3), (C3×C3⋊S3)⋊5S3, C3⋊S32(C3⋊S3), C325(C2×C3⋊S3), (C32×C3⋊S3)⋊5C2, (C3×C33⋊C2)⋊3C2, SmallGroup(324,170)

Series: Derived Chief Lower central Upper central

C1C34 — C3317D6
C1C3C32C33C34C3×C33⋊C2 — C3317D6
C34 — C3317D6
C1

Generators and relations for C3317D6
 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 [×3], C3 [×6], C3 [×13], C22, S3 [×21], C6 [×6], C32 [×2], C32 [×8], C32 [×40], D6 [×6], C3×S3 [×30], C3⋊S3, C3⋊S3 [×18], C3×C6, C33 [×6], C33 [×13], S32 [×9], C2×C3⋊S3, S3×C32 [×3], C3×C3⋊S3 [×4], C3×C3⋊S3 [×18], C33⋊C2 [×2], C34, S3×C3⋊S3 [×2], C324D6 [×4], C32×C3⋊S3, C3×C33⋊C2 [×2], C3317D6
Quotients: C1, C2 [×3], C22, S3 [×6], D6 [×6], C3⋊S3, S32 [×9], C2×C3⋊S3, S3×C3⋊S3 [×2], C324D6 [×4], C3317D6

Smallest permutation representation of C3317D6
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+++++++
imageC1C2C2S3S3D6S32C324D6
kernelC3317D6C32×C3⋊S3C3×C33⋊C2C3×C3⋊S3C33⋊C2C33C32C3
# reps11242698

Matrix representation of C3317D6 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] >;

C3317D6 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

׿
×
𝔽