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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 [×2], C3, C3 [×3], C4, C4, C22, S3 [×4], C6, C6 [×5], C2×C4, C32 [×2], C32, Dic3 [×2], C12, C12 [×4], D6 [×2], C2×C6, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, C4×S3 [×2], C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C3×C12, S3×C6, C2×C3⋊S3, C32⋊C6 [×2], C2×He3, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C4×C32⋊C6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C12 [×2], 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 10 6 9)(2 11 4 7)(3 12 5 8)(13 21 30 31)(14 22 25 32)(15 23 26 33)(16 24 27 34)(17 19 28 35)(18 20 29 36)

(2 28 25)(3 29 26)(4 17 14)(5 18 15)(7 19 22)(8 20 23)(11 35 32)(12 36 33)

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

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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