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G = C22×C32⋊D6  order 432 = 24·33

Direct product of C22 and C32⋊D6

direct product, non-abelian, supersoluble, monomial, rational

Aliases: C22×C32⋊D6, He3⋊C24, C62⋊6D6, C32⋊C6⋊C23, (C2×He3)⋊C23, C32⋊(S3×C23), He3⋊C2⋊C23, (C22×He3)⋊5C22, C6.93(C2×S32), (C2×C6).62S32, (C2×C3⋊S3)⋊7D6, C3⋊S3⋊(C22×S3), (C3×C6)⋊(C22×S3), C3.2(C22×S32), (C22×C3⋊S3)⋊5S3, (C22×C32⋊C6)⋊7C2, (C2×C32⋊C6)⋊10C22, (C22×He3⋊C2)⋊6C2, (C2×He3⋊C2)⋊8C22, SmallGroup(432,545)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — C22×C32⋊D6
Chief series
C1 — C3 — C32 — He3 — C32⋊C6 — C32⋊D6 — C2×C32⋊D6 — C22×C32⋊D6
Lower central
He3 — C22×C32⋊D6
Upper central
C1 — C22

Generators and relations for C22×C32⋊D6
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf=c-1d-1, ede-1=d-1, df=fd, fef=e-1 >

Subgroups: 3107 in 501 conjugacy classes, 109 normal (8 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C32, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×C6, He3, S32, S3×C6, C2×C3⋊S3, C62, C62, S3×C23, C32⋊C6, He3⋊C2, C2×He3, C2×S32, S3×C2×C6, C22×C3⋊S3, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, C22×He3, C22×S32, C2×C32⋊D6, C22×C32⋊C6, C22×He3⋊C2, C22×C32⋊D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, S32, S3×C23, C2×S32, C32⋊D6, C22×S32, C2×C32⋊D6, C22×C32⋊D6

Smallest permutation representation of C22×C32⋊D6
►On 36 points
Generators in S36
(1 10)(2 11)(3 12)(4 9)(5 7)(6 8)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)

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

(1 28 25)(2 29 26)(4 18 15)(6 17 14)(8 24 21)(9 19 22)(10 33 36)(11 34 31)

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

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

44 conjugacy classes

class 1 2A2B2C2D···2O3A3B3C3D6A6B6C6D···6I6J6K6L6M···6X
order12222···233336666···66666···6
size11119···9266122226···612121218···18

44 irreducible representations

dim11112224466
type+++++++++++
imageC1C2C2C2S3D6D6S32C2×S32C32⋊D6C2×C32⋊D6
kernelC22×C32⋊D6C2×C32⋊D6C22×C32⋊C6C22×He3⋊C2C22×C3⋊S3C2×C3⋊S3C62C2×C6C6C22C2
# reps1122121221326

Matrix representation of C22×C32⋊D6 ►in GL14(ℤ)

10000000000000
01000000000000
00100000000000
00010000000000
0000-1000000000
00000-100000000
000000-10000000
0000000-1000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
-10000000000000
0-1000000000000
00-100000000000
000-10000000000
00001000000000
00000100000000
00000010000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
0-1000000000000
1-1000000000000
000-10000000000
001-10000000000
00000-100000000
00001-100000000
0000000-1000000
0000001-1000000
00000000100000
00000000010000
0000000000-1100
0000000000-1000
00000000-1-10-1-1-1
00000000001010
,
10000000000000
01000000000000
00100000000000
00010000000000
00001000000000
00000100000000
00000010000000
00000001000000
00000000-110000
00000000-100000
0000000000-1100
0000000000-1000
00000000101001
000000000-10-1-1-1
,
0-10-10000000000
-10-100000000000
01000000000000
10000000000000
00000-10-1000000
0000-10-10000000
00000100000000
00001000000000
00000000-1-1-1-1-2-1
000000000000-11
00000000010000
00000000100000
00000000000010
00000000000110
,
01010000000000
10100000000000
000-10000000000
00-100000000000
00000-10-1000000
0000-10-10000000
00000001000000
00000010000000
000000000000-11
00000000-1-1-1-1-2-1
00000000001000
00000000000100
00000000000010
00000000100010

G:=sub<GL(14,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-2,-1,0,0,1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,-2,0,0,1,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0] >;
 

C22×C32⋊D6 in GAP, Magma, Sage, TeX 

C_2^2\times C_3^2\rtimes D_6
 
% in TeX
 
G:=Group("C2^2xC3^2:D6");
 
// GroupNames label
 
G:=SmallGroup(432,545);
 
// by ID
 
G=gap.SmallGroup(432,545);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^6=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=f*c*f=c^-1*d^-1,e*d*e^-1=d^-1,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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