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## G = C3×S3×D12order 432 = 24·33

### Direct product of C3, S3 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×S3×D12
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — S32×C6 — C3×S3×D12
 Lower central C32 — C3×C6 — C3×S3×D12
 Upper central C1 — C6 — C12

Generators and relations for C3×S3×D12
G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1096 in 270 conjugacy classes, 72 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×D12, S3×D4, C6×D4, S3×C32, S3×C32, C3×C3⋊S3, C32×C6, C3⋊D12, S3×C12, S3×C12, C3×D12, C3×D12, C3×C3⋊D4, C12⋊S3, C6×C12, D4×C32, C2×S32, S3×C2×C6, C32×Dic3, C32×C12, C3×S32, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, S3×D12, C6×D12, C3×S3×D4, C3×C3⋊D12, S3×C3×C12, C32×D12, C3×C12⋊S3, S32×C6, C3×S3×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, S32, S3×C6, C2×D12, S3×D4, C6×D4, C3×D12, C2×S32, S3×C2×C6, C3×S32, S3×D12, C6×D12, C3×S3×D4, S32×C6, C3×S3×D12

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

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 6P ··· 6Y 6Z ··· 6AE 6AF 6AG 6AH 6AI 12A ··· 12H 12I ··· 12Q 12R ··· 12Y 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 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 3 3 6 6 18 18 1 1 2 ··· 2 4 4 4 2 6 1 1 2 ··· 2 3 3 3 3 4 4 4 6 ··· 6 12 ··· 12 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

81 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 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 S3 D4 D6 D6 D6 C3×S3 C3×S3 D12 C3×D4 S3×C6 S3×C6 S3×C6 C3×D12 S32 S3×D4 C2×S32 C3×S32 S3×D12 C3×S3×D4 S32×C6 C3×S3×D12 kernel C3×S3×D12 C3×C3⋊D12 S3×C3×C12 C32×D12 C3×C12⋊S3 S32×C6 S3×D12 C3⋊D12 S3×C12 C3×D12 C12⋊S3 C2×S32 S3×C12 C3×D12 S3×C32 C3×Dic3 C3×C12 S3×C6 C4×S3 D12 C3×S3 C3×S3 Dic3 C12 D6 S3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 1 1 1 2 2 4 2 2 2 4 1 1 2 1 2 3 2 2 4 4 2 4 6 8 1 1 1 2 2 2 2 4

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

 9 0 0 0 0 0 0 9 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 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 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1

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

C3×S3×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C3xS3xD12");
// GroupNames label

G:=SmallGroup(432,649);
// by ID

G=gap.SmallGroup(432,649);
# by ID

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

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

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