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G = C12.86S32  order 432 = 24·33

6th non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

non-abelian, supersoluble, monomial

Aliases: C12.86S32, (C3×C12)⋊2D6, He3⋊3(C2×D4), C12⋊S3⋊6S3, C32⋊2(S3×D4), He3⋊C2⋊2D4, He3⋊2D4⋊2C2, He3⋊4D4⋊7C2, C4⋊2(C32⋊D6), (C4×He3)⋊2C22, C3.2(D6⋊D6), He3⋊3C4⋊2C22, (C2×He3).9C23, C6.83(C2×S32), (C2×C3⋊S3)⋊2D6, (C2×C32⋊D6)⋊2C2, (C4×He3⋊C2)⋊3C2, (C3×C6).9(C22×S3), C2.11(C2×C32⋊D6), (C2×C32⋊C6)⋊2C22, (C2×He3⋊C2).15C22, SmallGroup(432,302)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C12.86S32
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C2×C32⋊D6 — C12.86S32
Lower central
He3 — C2×He3 — C12.86S32
Upper central
C1 — C2 — C4

Generators and relations for C12.86S32
 G = < a,b,c,d | a3=b12=c6=d2=1, ab=ba, cac-1=dad=a-1b4, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 1555 in 205 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, He3, C3×Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×D12, S3×D4, C32⋊C6, He3⋊C2, C2×He3, C3⋊D12, S3×C12, C3×D12, C12⋊S3, C2×S32, He3⋊3C4, C4×He3, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, S3×D12, He3⋊2D4, He3⋊4D4, C4×He3⋊C2, C2×C32⋊D6, C12.86S32
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, S32, S3×D4, C2×S32, C32⋊D6, D6⋊D6, C2×C32⋊D6, C12.86S32

Smallest permutation representation of C12.86S32
►On 36 points
Generators in S36
(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E12F12G12H
order1222222233334466666666661212121212121212
size119918181818266122182661218183636363622121212121818

32 irreducible representations

dim1111122224444666
type+++++++++++++++
imageC1C2C2C2C2S3D4D6D6S32S3×D4C2×S32D6⋊D6C32⋊D6C2×C32⋊D6C12.86S32
kernelC12.86S32He3⋊2D4He3⋊4D4C4×He3⋊C2C2×C32⋊D6C12⋊S3He3⋊C2C3×C12C2×C3⋊S3C12C32C6C3C4C2C1
# reps1221222241212224

Matrix representation of C12.86S32 ►in GL6(𝔽13)

100000
010000
0001200
0011200
0000121
0000120
,
330000
1060000
003300
0010600
000033
0000106
,
0000012
0000120
0120000
1200000
0001200
0012000
,
000010
000001
001000
000100
100000
010000

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

C12.86S32 in GAP, Magma, Sage, TeX 

C_{12}._{86}S_3^2
 
% in TeX
 
G:=Group("C12.86S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,302);
 
// by ID
 
G=gap.SmallGroup(432,302);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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