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G = C62⋊D6  order 432 = 24·33

1st semidirect product of C62 and D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C62⋊1D6, He3⋊6(C2×D4), C3⋊Dic3⋊1D6, C32⋊C6⋊2D4, C32⋊3(S3×D4), He3⋊3D4⋊5C2, He3⋊6D4⋊3C2, He3⋊2D4⋊5C2, He3⋊7D4⋊2C2, C32⋊7D4⋊3S3, He3⋊3C4⋊C22, C32⋊C12⋊1C22, C22⋊2(C32⋊D6), (C2×He3).17C23, (C22×He3)⋊1C22, (C2×C6).9S32, C3⋊S3⋊(C3⋊D4), C6.91(C2×S32), (C2×C3⋊S3)⋊6D6, C3.2(S3×C3⋊D4), C6.S32⋊5C2, (C22×C3⋊S3)⋊3S3, (C2×C32⋊D6)⋊3C2, C32⋊3(C2×C3⋊D4), C2.17(C2×C32⋊D6), (C22×C32⋊C6)⋊3C2, (C2×C32⋊C6)⋊6C22, (C3×C6).17(C22×S3), (C2×He3⋊C2)⋊2C22, SmallGroup(432,323)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C62⋊D6
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C2×C32⋊D6 — C62⋊D6
Lower central
He3 — C2×He3 — C62⋊D6
Upper central
C1 — C2 — C22

Generators and relations for C62⋊D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 1419 in 205 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, S3×D4, C2×C3⋊D4, C32⋊C6, C32⋊C6, He3⋊C2, C2×He3, C2×He3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C32⋊C12, He3⋊3C4, C32⋊D6, C2×C32⋊C6, C2×C32⋊C6, C2×He3⋊C2, C22×He3, S3×C3⋊D4, Dic3⋊D6, C6.S32, He3⋊2D4, He3⋊3D4, He3⋊6D4, He3⋊7D4, C2×C32⋊D6, C22×C32⋊C6, C62⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, S32, S3×D4, C2×C3⋊D4, C2×S32, C32⋊D6, S3×C3⋊D4, C2×C32⋊D6, C62⋊D6

Smallest permutation representation of C62⋊D6
►On 36 points
Generators in S36
(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 5 3)(7 12 9 8 11 10)(13 23 14 24 15 22)(16 19 18 21 17 20)(25 26 27 28 29 30)(31 36 35 34 33 32)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P12A12B
order1222222233334466666666666666661212
size11299181818266121818246666121212121818181836363636

32 irreducible representations

dim11111111122222222444466
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C62⋊D6S3S3D4D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4C32⋊D6C2×C32⋊D6
kernelC62⋊D6C6.S32He3⋊2D4He3⋊3D4He3⋊6D4He3⋊7D4C2×C32⋊D6C22×C32⋊C6C1C32⋊7D4C22×C3⋊S3C32⋊C6C3⋊Dic3C2×C3⋊S3C62C3⋊S3C2×C6C32C6C3C22C2
# reps1111111111121324111222

Matrix representation of C62⋊D6 ►in GL10(ℤ)

1000000000
0100000000
00-10000000
000-1000000
0000100000
0000010000
0000000100
000000-1-100
00000000-1-1
0000000010
,
-1000000000
0-100000000
00-10000000
000-1000000
0000-1-10000
0000100000
000000-1-100
0000001000
00000000-1-1
0000000010
,
00-1-1000000
0010000000
-1-100000000
1000000000
00000000-10
0000000011
0000-100000
0000110000
000000-1000
0000001100
,
00-1-1000000
0001000000
-1-100000000
0100000000
00000000-10
000000000-1
000000-1000
0000000-100
0000-100000
00000-10000

G:=sub<GL(10,Integers())| [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,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0],[-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,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0],[0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0] >;
 

C62⋊D6 in GAP, Magma, Sage, TeX 

C_6^2\rtimes D_6
 
% in TeX
 
G:=Group("C6^2:D6");
 
// GroupNames label
 
G:=SmallGroup(432,323);
 
// by ID
 
G=gap.SmallGroup(432,323);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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