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G = C6xC6.D6order 432 = 24·33

Direct product of C6 and C6.D6

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

Aliases: C6xC6.D6, C62.109D6, C6:1(S3xC12), (C6xDic3):8C6, Dic3:6(S3xC6), (C3xDic3):20D6, (C6xDic3):11S3, C62.25(C2xC6), C33:11(C22xC4), C32:6(C22xC12), (C32xC6).32C23, (C3xC62).19C22, (C32xDic3):18C22, C2.3(S32xC6), C3:2(S3xC2xC12), (C6xC3:S3):7C4, (C3xC6):8(C4xS3), (C2xC6).72S32, C6.13(S3xC2xC6), (C2xC3:S3):6C12, C3:S3:3(C2xC12), C6.116(C2xS32), (C3xC6):5(C2xC12), C32:16(S3xC2xC4), C22.9(C3xS32), (C2xC6).27(S3xC6), (C32xC6):5(C2xC4), (Dic3xC3xC6):12C2, (C3xDic3):7(C2xC6), (C2xDic3):5(C3xS3), (C22xC3:S3).7C6, (C6xC3:S3).49C22, (C3xC6).23(C22xC6), (C3xC6).137(C22xS3), (C2xC6xC3:S3).7C2, (C3xC3:S3):8(C2xC4), (C2xC3:S3).21(C2xC6), SmallGroup(432,654)

Series: Derived Chief Lower central Upper central

C1C32 — C6xC6.D6
C1C3C32C3xC6C32xC6C32xDic3C3xC6.D6 — C6xC6.D6
C32 — C6xC6.D6
C1C2xC6

Generators and relations for C6xC6.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, C2xC4, C23, C32, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22xC4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C22xC6, C33, C3xDic3, C3xDic3, C3xC12, S3xC6, C2xC3:S3, C62, C62, C62, S3xC2xC4, C22xC12, C3xC3:S3, C32xC6, C32xC6, C6.D6, S3xC12, C6xDic3, C6xDic3, C6xC12, S3xC2xC6, C22xC3:S3, C32xDic3, C6xC3:S3, C3xC62, C2xC6.D6, S3xC2xC12, C3xC6.D6, Dic3xC3xC6, C2xC6xC3:S3, C6xC6.D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, C12, D6, C2xC6, C22xC4, C3xS3, C4xS3, C2xC12, C22xS3, C22xC6, S32, S3xC6, S3xC2xC4, C22xC12, C6.D6, S3xC12, C2xS32, S3xC2xC6, C3xS32, C2xC6.D6, S3xC2xC12, C3xC6.D6, S32xC6, C6xC6.D6

Smallest permutation representation of C6xC6.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++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6D6C3xS3C4xS3S3xC6S3xC6S3xC12S32C6.D6C2xS32C3xS32C3xC6.D6S32xC6
kernelC6xC6.D6C3xC6.D6Dic3xC3xC6C2xC6xC3:S3C2xC6.D6C6xC3:S3C6.D6C6xDic3C22xC3:S3C2xC3:S3C6xDic3C3xDic3C62C2xDic3C3xC6Dic3C2xC6C6C2xC6C6C6C22C2C2
# reps14212884216242488416121242

Matrix representation of C6xC6.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] >;

C6xC6.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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