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## G = C3×D12⋊S3order 432 = 24·33

### Direct product of C3 and D12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3×D12⋊S3
 Lower central C32 — C3×C6 — C3×D12⋊S3
 Upper central C1 — C6 — C12

Generators and relations for C3×D12⋊S3
G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 704 in 210 conjugacy classes, 64 normal (40 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, D42S3, Q83S3, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, C3⋊D12, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C3×D12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, D4×C32, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C32×C12, S3×C3×C6, C6×C3⋊S3, D12⋊S3, C3×D42S3, C3×Q83S3, C3×S3×Dic3, C3×C3⋊D12, C32×Dic6, C32×D12, C12×C3⋊S3, C3×D12⋊S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, D42S3, Q83S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D12⋊S3, C3×D42S3, C3×Q83S3, S32×C6, C3×D12⋊S3

Smallest permutation representation of C3×D12⋊S3
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 38)(2 43)(3 48)(4 41)(5 46)(6 39)(7 44)(8 37)(9 42)(10 47)(11 40)(12 45)(13 26)(14 31)(15 36)(16 29)(17 34)(18 27)(19 32)(20 25)(21 30)(22 35)(23 28)(24 33)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 6P ··· 6U 6V 6W 12A 12B 12C ··· 12N 12O 12P 12Q 12R 12S 12T 12U 12V 12W ··· 12AB order 1 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 12 12 12 12 12 12 12 ··· 12 size 1 1 6 6 18 1 1 2 ··· 2 4 4 4 2 6 6 9 9 1 1 2 ··· 2 4 4 4 6 6 6 6 12 ··· 12 18 18 2 2 4 ··· 4 6 6 6 6 9 9 9 9 12 ··· 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 S3 D6 D6 D6 C4○D4 C3×S3 C3×S3 S3×C6 S3×C6 S3×C6 C3×C4○D4 S32 D4⋊2S3 Q8⋊3S3 C2×S32 C3×S32 D12⋊S3 C3×D4⋊2S3 C3×Q8⋊3S3 S32×C6 C3×D12⋊S3 kernel C3×D12⋊S3 C3×S3×Dic3 C3×C3⋊D12 C32×Dic6 C32×D12 C12×C3⋊S3 D12⋊S3 S3×Dic3 C3⋊D12 C3×Dic6 C3×D12 C4×C3⋊S3 C3×Dic6 C3×D12 C3×Dic3 C3×C12 S3×C6 C33 Dic6 D12 Dic3 C12 D6 C32 C12 C32 C32 C6 C4 C3 C3 C3 C2 C1 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 1 1 1 1 2 2 2 2 2 4

Matrix representation of C3×D12⋊S3 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 5 5 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1 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
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×D12⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_{12}\rtimes S_3
% in TeX

G:=Group("C3xD12:S3");
// GroupNames label

G:=SmallGroup(432,644);
// by ID

G=gap.SmallGroup(432,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=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,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations

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