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## G = C6×C6.D6order 432 = 24·33

### Direct product of C6 and C6.D6

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 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A ··· 4H 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 6AH ··· 6AO 12A ··· 12P 12Q ··· 12AN order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 1 1 2 ··· 2 4 4 4 3 ··· 3 1 ··· 1 2 ··· 2 4 ··· 4 9 ··· 9 3 ··· 3 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 S32 C6.D6 C2×S32 C3×S32 C3×C6.D6 S32×C6 kernel C6×C6.D6 C3×C6.D6 Dic3×C3×C6 C2×C6×C3⋊S3 C2×C6.D6 C6×C3⋊S3 C6.D6 C6×Dic3 C22×C3⋊S3 C2×C3⋊S3 C6×Dic3 C3×Dic3 C62 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 2 1 2 8 8 4 2 16 2 4 2 4 8 8 4 16 1 2 1 2 4 2

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

 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 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 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 5 0 0 0 0 8 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 1 0 0 0 0 0 1

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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