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G = C6×C6.D6  order 432 = 24·33

Direct product of C6 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C6×C6.D6, C62.109D6, C6⋊1(S3×C12), (C6×Dic3)⋊8C6, Dic3⋊6(S3×C6), (C3×Dic3)⋊20D6, (C6×Dic3)⋊11S3, C62.25(C2×C6), C33⋊11(C22×C4), C32⋊6(C22×C12), (C32×C6).32C23, (C3×C62).19C22, (C32×Dic3)⋊18C22, C2.3(S32×C6), C3⋊2(S3×C2×C12), (C6×C3⋊S3)⋊7C4, (C3×C6)⋊8(C4×S3), (C2×C6).72S32, C6.13(S3×C2×C6), (C2×C3⋊S3)⋊6C12, C3⋊S3⋊3(C2×C12), C6.116(C2×S32), (C3×C6)⋊5(C2×C12), C32⋊16(S3×C2×C4), C22.9(C3×S32), (C2×C6).27(S3×C6), (C32×C6)⋊5(C2×C4), (Dic3×C3×C6)⋊12C2, (C3×Dic3)⋊7(C2×C6), (C2×Dic3)⋊5(C3×S3), (C22×C3⋊S3).7C6, (C6×C3⋊S3).49C22, (C3×C6).23(C22×C6), (C3×C6).137(C22×S3), (C2×C6×C3⋊S3).7C2, (C3×C3⋊S3)⋊8(C2×C4), (C2×C3⋊S3).21(C2×C6), SmallGroup(432,654)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C6×C6.D6
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — C32×Dic3 — C3×C6.D6 — C6×C6.D6
Lower central
C32 — C6×C6.D6
Upper central
C1 — C2×C6

Generators and relations for C6×C6.D6
 G = < a,b,c,d | a6=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 928 in 290 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C22×C12, C3×C3⋊S3, C32×C6, C32×C6, C6.D6, S3×C12, C6×Dic3, C6×Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C6×C3⋊S3, C3×C62, C2×C6.D6, S3×C2×C12, C3×C6.D6, Dic3×C3×C6, C2×C6×C3⋊S3, C6×C6.D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S32, S3×C6, S3×C2×C4, C22×C12, C6.D6, S3×C12, C2×S32, S3×C2×C6, C3×S32, C2×C6.D6, S3×C2×C12, C3×C6.D6, S32×C6, C6×C6.D6

Smallest permutation representation of C6×C6.D6
►On 48 points
Generators in S48
(1 23 5 15 9 19)(2 24 6 16 10 20)(3 13 7 17 11 21)(4 14 8 18 12 22)(25 45 33 41 29 37)(26 46 34 42 30 38)(27 47 35 43 31 39)(28 48 36 44 32 40)

(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 48 46 44 42 40)

(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)(37 38 39 40 41 42 43 44 45 46 47 48)

(1 48)(2 41)(3 46)(4 39)(5 44)(6 37)(7 42)(8 47)(9 40)(10 45)(11 38)(12 43)(13 34)(14 27)(15 32)(16 25)(17 30)(18 35)(19 28)(20 33)(21 26)(22 31)(23 36)(24 29)
 
G:=sub<Sym(48)| (1,23,5,15,9,19)(2,24,6,16,10,20)(3,13,7,17,11,21)(4,14,8,18,12,22)(25,45,33,41,29,37)(26,46,34,42,30,38)(27,47,35,43,31,39)(28,48,36,44,32,40), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,48)(2,41)(3,46)(4,39)(5,44)(6,37)(7,42)(8,47)(9,40)(10,45)(11,38)(12,43)(13,34)(14,27)(15,32)(16,25)(17,30)(18,35)(19,28)(20,33)(21,26)(22,31)(23,36)(24,29)>;
 
G:=Group( (1,23,5,15,9,19)(2,24,6,16,10,20)(3,13,7,17,11,21)(4,14,8,18,12,22)(25,45,33,41,29,37)(26,46,34,42,30,38)(27,47,35,43,31,39)(28,48,36,44,32,40), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,48)(2,41)(3,46)(4,39)(5,44)(6,37)(7,42)(8,47)(9,40)(10,45)(11,38)(12,43)(13,34)(14,27)(15,32)(16,25)(17,30)(18,35)(19,28)(20,33)(21,26)(22,31)(23,36)(24,29) );
 
G=PermutationGroup([[(1,23,5,15,9,19),(2,24,6,16,10,20),(3,13,7,17,11,21),(4,14,8,18,12,22),(25,45,33,41,29,37),(26,46,34,42,30,38),(27,47,35,43,31,39),(28,48,36,44,32,40)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24),(25,27,29,31,33,35),(26,36,34,32,30,28),(37,39,41,43,45,47),(38,48,46,44,42,40)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,48),(2,41),(3,46),(4,39),(5,44),(6,37),(7,42),(8,47),(9,40),(10,45),(11,38),(12,43),(13,34),(14,27),(15,32),(16,25),(17,30),(18,35),(19,28),(20,33),(21,26),(22,31),(23,36),(24,29)]])
 

108 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H3I3J3K4A···4H6A···6F6G···6X6Y···6AG6AH···6AO12A···12P12Q···12AN
order12222222333···33334···46···66···66···66···612···1212···12
size11119999112···24443···31···12···24···49···93···36···6

108 irreducible representations

dim111111111122222222444444
type++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6D6C3×S3C4×S3S3×C6S3×C6S3×C12S32C6.D6C2×S32C3×S32C3×C6.D6S32×C6
kernelC6×C6.D6C3×C6.D6Dic3×C3×C6C2×C6×C3⋊S3C2×C6.D6C6×C3⋊S3C6.D6C6×Dic3C22×C3⋊S3C2×C3⋊S3C6×Dic3C3×Dic3C62C2×Dic3C3×C6Dic3C2×C6C6C2×C6C6C6C22C2C2
# reps14212884216242488416121242

Matrix representation of C6×C6.D6 ►in GL6(𝔽13)

1200000
0120000
003000
000300
000010
000001
,
1210000
1200000
0012100
0012000
0000120
0000012
,
010000
100000
000100
001000
000005
000085
,
010000
100000
000100
001000
0000121
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
 

C6×C6.D6 in GAP, Magma, Sage, TeX 

C_6\times C_6.D_6
 
% in TeX
 
G:=Group("C6xC6.D6");
 
// GroupNames label
 
G:=SmallGroup(432,654);
 
// by ID
 
G=gap.SmallGroup(432,654);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,176,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
 
// generators/relations
 

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