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G = C6.S32  order 216 = 23·33

2nd non-split extension by C6 of S32 acting via S32/C3⋊S3=C2

non-abelian, supersoluble, monomial

Aliases: C6.17S32, C32⋊C6⋊C4, C3⋊S3⋊Dic3, He3⋊2(C2×C4), (C3×C6).2D6, C3⋊Dic3⋊1S3, C32⋊1(C4×S3), C32⋊(C2×Dic3), C32⋊C12⋊2C2, He3⋊3C4⋊2C2, C3.2(S3×Dic3), C2.1(C32⋊D6), (C2×He3).2C22, (C2×C3⋊S3).S3, (C2×C32⋊C6).C2, SmallGroup(216,34)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C6.S32
 G = < a,b,c,d | a3=b3=c6=d4=1, dad-1=ab=ba, cac-1=a-1b-1, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 282 in 62 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, S3×Dic3, C6.D6, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C6.S32
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C4×S3, C2×Dic3, S32, S3×Dic3, C32⋊D6, C6.S32

Character table of C6.S32

 class 12A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F12A12B12C12D
 size 119926612999926612181818181818
ρ11111111111111111111111    trivial
ρ211-1-111111-11-11111-1-111-1-1    linear of order 2
ρ311-1-11111-11-111111-1-1-1-111    linear of order 2
ρ411111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ51-1-111111-i-iii-1-1-1-11-1-ii-ii    linear of order 4
ρ61-11-11111-iii-i-1-1-1-1-11-iii-i    linear of order 4
ρ71-11-11111i-i-ii-1-1-1-1-11i-i-ii    linear of order 4
ρ81-1-111111ii-i-i-1-1-1-11-1i-ii-i    linear of order 4
ρ922002-12-10-20-222-1-1000011    orthogonal lifted from D6
ρ10222222-1-100002-12-1-1-10000    orthogonal lifted from S3
ρ1122002-12-1020222-1-10000-1-1    orthogonal lifted from S3
ρ1222-2-222-1-100002-12-1110000    orthogonal lifted from D6
ρ132-2-2222-1-10000-21-21-110000    symplectic lifted from Dic3, Schur index 2
ρ142-22-222-1-10000-21-211-10000    symplectic lifted from Dic3, Schur index 2
ρ152-2002-12-102i0-2i-2-2110000-ii    complex lifted from C4×S3
ρ162-2002-12-10-2i02i-2-2110000i-i    complex lifted from C4×S3
ρ1744004-2-2100004-2-21000000    orthogonal lifted from S32
ρ184-4004-2-210000-422-1000000    symplectic lifted from S3×Dic3, Schur index 2
ρ196600-3000-20-20-3000001100    orthogonal lifted from C32⋊D6
ρ206600-30002020-300000-1-100    orthogonal lifted from C32⋊D6
ρ216-600-3000-2i02i0300000i-i00    complex faithful
ρ226-600-30002i0-2i0300000-ii00    complex faithful

Smallest permutation representation of C6.S32
►On 36 points
Generators in S36
(1 28 19)(2 29 20)(4 22 25)(5 23 26)(8 32 17)(9 33 18)(11 14 35)(12 15 36)

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

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

C6.S32 is a maximal subgroup of   C3⋊S3⋊Dic6  C12⋊S3⋊S3  C12.84S32  C4×C32⋊D6  C62.8D6  C62.9D6  C62⋊D6
C6.S32 is a maximal quotient of   C32⋊C6⋊C8  He3⋊M4(2)  He3⋊C42  C62.D6  C62.4D6

Matrix representation of C6.S32 ►in GL6(𝔽13)

1210000
1200000
001000
000100
0000012
0000112
,
1210000
1200000
0012100
0012000
0000121
0000120
,
0001200
0012000
0000012
0000120
0120000
1200000
,
080000
800000
000008
000080
000800
008000

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,0,0,0,0,12,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0] >;
 

C6.S32 in GAP, Magma, Sage, TeX 

C_6.S_3^2
 
% in TeX
 
G:=Group("C6.S3^2");
 
// GroupNames label
 
G:=SmallGroup(216,34);
 
// by ID
 
G=gap.SmallGroup(216,34);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,201,1444,382,5189,2603]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^6=d^4=1,d*a*d^-1=a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of C6.S32 in TeX

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