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G = C4×C32⋊C6  order 216 = 23·33

Direct product of C4 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C4×C32⋊C6, C3⋊S3⋊C12, (C3×C12)⋊3S3, (C3×C12)⋊2C6, He3⋊4(C2×C4), (C3×C6).7D6, C6.10(S3×C6), C3.2(S3×C12), (C4×He3)⋊3C2, C3⋊Dic3⋊2C6, C32⋊3(C4×S3), C12.12(C3×S3), C32⋊C12⋊5C2, C32⋊1(C2×C12), (C2×He3).7C22, (C4×C3⋊S3)⋊C3, (C2×C3⋊S3).C6, (C3×C6).2(C2×C6), C2.1(C2×C32⋊C6), (C2×C32⋊C6).2C2, SmallGroup(216,50)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C4×C32⋊C6
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6
Lower central
C32 — C4×C32⋊C6
Upper central
C1 — C4

Generators and relations for C4×C32⋊C6
 G = < a,b,c,d | a4=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 232 in 62 conjugacy classes, 25 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C4×C32⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, C4×S3, C2×C12, S3×C6, C32⋊C6, S3×C12, C2×C32⋊C6, C4×C32⋊C6

Smallest permutation representation of C4×C32⋊C6
►On 36 points
Generators in S36
(1 4 11 9)(2 5 12 7)(3 6 10 8)(13 28 19 35)(14 29 20 36)(15 30 21 31)(16 25 22 32)(17 26 23 33)(18 27 24 34)

(2 17 14)(3 18 15)(5 26 29)(6 27 30)(7 33 36)(8 34 31)(10 24 21)(12 23 20)

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

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

C4×C32⋊C6 is a maximal subgroup of
 C32⋊C6⋊C8  He3⋊M4(2)  He3⋊5M4(2)  C3⋊S3⋊Dic6  C12⋊S3⋊S3  C12.91S32  C12.S32  C3⋊S3⋊D12  C62.36D6  C62.13D6  (Q8×He3)⋊C2
C4×C32⋊C6 is a maximal quotient of
 He3⋊5M4(2)  C62.19D6  C62.21D6

40 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E12F12G···12L12M12N12O12P
order12223333334444666666666612121212121212···1212121212
size1199233666119923366699992233336···69999

40 irreducible representations

dim1111111111222222666
type++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12C32⋊C6C2×C32⋊C6C4×C32⋊C6
kernelC4×C32⋊C6C32⋊C12C4×He3C2×C32⋊C6C4×C3⋊S3C32⋊C6C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C3×C12C3×C6C12C32C6C3C4C2C1
# reps1111242228112224112

Matrix representation of C4×C32⋊C6 ►in GL8(𝔽13)

80000000
08000000
00100000
00010000
00001000
00000100
00000010
00000001
,
012000000
112000000
00100000
00010000
000012100
000012000
000000012
000000112
,
10000000
01000000
000120000
001120000
000001200
000011200
000000012
000000112
,
01000000
10000000
00000100
00001000
00000001
00000010
00010000
00100000

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

C4×C32⋊C6 in GAP, Magma, Sage, TeX 

C_4\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C4xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(216,50);
 
// by ID
 
G=gap.SmallGroup(216,50);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,1444,736,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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