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## G = C6×C3⋊D12order 432 = 24·33

### Direct product of C6 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — C6×C3⋊D12
 Lower central C32 — C3×C6 — C6×C3⋊D12
 Upper central C1 — C2×C6

Generators and relations for C6×C3⋊D12
G = < a,b,c,d | a6=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1120 in 306 conjugacy classes, 80 normal (36 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C2×D12, C2×C3⋊D4, C6×D4, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, C3⋊D12, C3×D12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C2×C62, C32×Dic3, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C2×C3⋊D12, C6×D12, C6×C3⋊D4, C3×C3⋊D12, Dic3×C3×C6, S3×C62, C2×C6×C3⋊S3, C6×C3⋊D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3⋊D4, C3×D4, C22×S3, C22×C6, S32, S3×C6, C2×D12, C2×C3⋊D4, C6×D4, C3⋊D12, C3×D12, C3×C3⋊D4, C2×S32, S3×C2×C6, C3×S32, C2×C3⋊D12, C6×D12, C6×C3⋊D4, C3×C3⋊D12, S32×C6, C6×C3⋊D12

Smallest permutation representation of C6×C3⋊D12
On 48 points
Generators in S48
(1 34 9 30 5 26)(2 35 10 31 6 27)(3 36 11 32 7 28)(4 25 12 33 8 29)(13 42 17 46 21 38)(14 43 18 47 22 39)(15 44 19 48 23 40)(16 45 20 37 24 41)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 45 41)(38 42 46)(39 47 43)(40 44 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)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)

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

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

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

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 ··· 6AW 6AX 6AY 6AZ 6BA 12A ··· 12P 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 6 6 6 6 12 ··· 12 size 1 1 1 1 6 6 18 18 1 1 2 ··· 2 4 4 4 6 6 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 S3 D4 D6 D6 D6 C3×S3 C3×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C6 S3×C6 C3×D12 C3×C3⋊D4 S32 C3⋊D12 C2×S32 C3×S32 C3×C3⋊D12 S32×C6 kernel C6×C3⋊D12 C3×C3⋊D12 Dic3×C3×C6 S3×C62 C2×C6×C3⋊S3 C2×C3⋊D12 C3⋊D12 C6×Dic3 S3×C2×C6 C22×C3⋊S3 C6×Dic3 S3×C2×C6 C32×C6 C3×Dic3 S3×C6 C62 C2×Dic3 C22×S3 C3×C6 C3×C6 C3×C6 Dic3 D6 C2×C6 C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 2 8 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 1 2 1 2 4 2

Matrix representation of C6×C3⋊D12 in GL6(𝔽13)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 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
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 11 6 0 0 0 0 6 2 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,6,0,0,0,0,6,2,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6×C3⋊D12 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes D_{12}
% in TeX

G:=Group("C6xC3:D12");
// GroupNames label

G:=SmallGroup(432,656);
// by ID

G=gap.SmallGroup(432,656);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,176,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^12=d^2=1,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^-1>;
// generators/relations

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