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G = C4×C32⋊D6  order 432 = 24·33

Direct product of C4 and C32⋊D6

direct product, non-abelian, supersoluble, monomial

Aliases: C4×C32⋊D6, C12.92S32, (C3×C12)⋊5D6, C3⋊Dic3⋊3D6, He3⋊1(C22×C4), (C4×He3)⋊5C22, C32⋊C12⋊5C22, He3⋊3C4⋊4C22, (C2×He3).7C23, C3⋊S3⋊(C4×S3), C3.2(C4×S32), (C4×C3⋊S3)⋊5S3, C6.81(C2×S32), C32⋊1(S3×C2×C4), (C2×C3⋊S3).8D6, C6.S32⋊6C2, He3⋊(C2×C4)⋊5C2, (C4×C32⋊C6)⋊8C2, C32⋊C6⋊1(C2×C4), C2.1(C2×C32⋊D6), (C4×He3⋊C2)⋊7C2, He3⋊C2⋊1(C2×C4), (C3×C6).7(C22×S3), (C2×C32⋊D6).2C2, (C2×C32⋊C6).8C22, (C2×He3⋊C2).14C22, SmallGroup(432,300)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C4×C32⋊D6
 G = < a,b,c,d,e | a4=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1c-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 1171 in 205 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, C32⋊C6, He3⋊C2, C2×He3, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C32⋊C12, He3⋊3C4, C4×He3, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, C4×S32, C6.S32, He3⋊(C2×C4), C4×C32⋊C6, C4×He3⋊C2, C2×C32⋊D6, C4×C32⋊D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C2×S32, C32⋊D6, C4×S32, C2×C32⋊D6, C4×C32⋊D6

Smallest permutation representation of C4×C32⋊D6
►On 36 points
Generators in S36
(1 8 10 6)(2 9 11 4)(3 7 12 5)(13 29 19 35)(14 30 20 36)(15 25 21 31)(16 26 22 32)(17 27 23 33)(18 28 24 34)

(1 24 21)(3 23 20)(5 27 30)(6 28 25)(7 33 36)(8 34 31)(10 18 15)(12 17 14)

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

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

44 conjugacy classes

class 1 2A2B···2G3A3B3C3D4A4B4C···4H6A6B6C6D6E···6J12A12B12C12D12E12F12G12H12I···12N
order122···23333444···466666···6121212121212121212···12
size119···926612119···92661218···18226666121218···18

44 irreducible representations

dim111111122222444666
type++++++++++++++
imageC1C2C2C2C2C2C4S3D6D6D6C4×S3S32C2×S32C4×S32C32⋊D6C2×C32⋊D6C4×C32⋊D6
kernelC4×C32⋊D6C6.S32He3⋊(C2×C4)C4×C32⋊C6C4×He3⋊C2C2×C32⋊D6C32⋊D6C4×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C12C6C3C4C2C1
# reps121211822228112224

Matrix representation of C4×C32⋊D6 ►in GL10(𝔽13)

8000000000
0800000000
0080000000
0008000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
01200000000
11200000000
00012000000
00112000000
0000100000
0000010000
0000000100
000011121200
000011001212
0000000010
,
1000000000
0100000000
0010000000
0001000000
00000120000
00001120000
000010121200
00000121000
000010001212
00000120010
,
01201000000
12010000000
01200000000
12000000000
000011111200
00000012100
00000012010
0000111201212
00000012000
00000112000
,
01200000000
12000000000
01201000000
12010000000
00000012100
000011111200
00000012000
00001012000
00000012010
00000012001

G:=sub<GL(10,GF(13))| [8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,12,12,0,12,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[0,12,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,11,12,12,12,12,12,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0],[0,12,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,11,12,12,12,12,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1] >;
 

C4×C32⋊D6 in GAP, Magma, Sage, TeX 

C_4\times C_3^2\rtimes D_6
 
% in TeX
 
G:=Group("C4xC3^2:D6");
 
// GroupNames label
 
G:=SmallGroup(432,300);
 
// by ID
 
G=gap.SmallGroup(432,300);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1*c^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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