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## G = C6×D6⋊S3order 432 = 24·33

### Direct product of C6 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×D6⋊S3 — C6×D6⋊S3
 Lower central C32 — C3×C6 — C6×D6⋊S3
 Upper central C1 — C2×C6

Generators and relations for C6×D6⋊S3
G = < a,b,c,d,e | a6=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 960 in 306 conjugacy classes, 80 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, S3×C32, C32×C6, C32×C6, D6⋊S3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, S3×C2×C6, C2×C62, C3×C3⋊Dic3, S3×C3×C6, S3×C3×C6, C3×C62, C2×D6⋊S3, C6×C3⋊D4, C3×D6⋊S3, C6×C3⋊Dic3, S3×C62, C6×D6⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S32, S3×C6, C2×C3⋊D4, C6×D4, D6⋊S3, C3×C3⋊D4, C2×S32, S3×C2×C6, C3×S32, C2×D6⋊S3, C6×C3⋊D4, C3×D6⋊S3, S32×C6, C6×D6⋊S3

Smallest permutation representation of C6×D6⋊S3
On 48 points
Generators in S48
(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 18 5 16 3 14)(2 13 6 17 4 15)(7 45 9 47 11 43)(8 46 10 48 12 44)(19 27 21 29 23 25)(20 28 22 30 24 26)(31 37 35 41 33 39)(32 38 36 42 34 40)
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 45)(32 46)(33 47)(34 48)(35 43)(36 44)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 30)(8 25)(9 26)(10 27)(11 28)(12 29)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 6AH ··· 6BM 12A 12B 12C 12D 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 6 6 6 6 1 1 2 ··· 2 4 4 4 18 18 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C3⋊D4 S32 D6⋊S3 C2×S32 C3×S32 C3×D6⋊S3 S32×C6 kernel C6×D6⋊S3 C3×D6⋊S3 C6×C3⋊Dic3 S3×C62 C2×D6⋊S3 D6⋊S3 C2×C3⋊Dic3 S3×C2×C6 S3×C2×C6 C32×C6 S3×C6 C62 C22×S3 C3×C6 C3×C6 D6 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 2 2 8 2 4 2 2 4 2 4 8 4 8 4 16 1 2 1 2 4 2

Matrix representation of C6×D6⋊S3 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 1 3 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 4 0 0 0 0 6 9 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,6,0,0,0,0,4,9,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C6×D6⋊S3 in GAP, Magma, Sage, TeX

C_6\times D_6\rtimes S_3
% in TeX

G:=Group("C6xD6:S3");
// GroupNames label

G:=SmallGroup(432,655);
// by ID

G=gap.SmallGroup(432,655);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

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