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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 — C32×C6 — C3⋊S3×D12
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — C3⋊S3×D12
 Lower central C33 — C32×C6 — C3⋊S3×D12
 Upper central C1 — C2 — C4

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

Subgroups: 2904 in 388 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, S3×D4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C3⋊D12, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×S32, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, S3×D12, D4×C3⋊S3, C337D4, C32×D12, C12×C3⋊S3, C3312D4, C2×S3×C3⋊S3, C3⋊S3×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, S32, C2×C3⋊S3, C2×D12, S3×D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×D12, D4×C3⋊S3, C2×S3×C3⋊S3, C3⋊S3×D12

Smallest permutation representation of C3⋊S3×D12
On 72 points
Generators in S72
(1 19 42)(2 20 43)(3 21 44)(4 22 45)(5 23 46)(6 24 47)(7 13 48)(8 14 37)(9 15 38)(10 16 39)(11 17 40)(12 18 41)(25 62 51)(26 63 52)(27 64 53)(28 65 54)(29 66 55)(30 67 56)(31 68 57)(32 69 58)(33 70 59)(34 71 60)(35 72 49)(36 61 50)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 63)(2 62)(3 61)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 60)(24 59)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 48)(33 47)(34 46)(35 45)(36 44)

G:=sub<Sym(72)| (1,19,42)(2,20,43)(3,21,44)(4,22,45)(5,23,46)(6,24,47)(7,13,48)(8,14,37)(9,15,38)(10,16,39)(11,17,40)(12,18,41)(25,62,51)(26,63,52)(27,64,53)(28,65,54)(29,66,55)(30,67,56)(31,68,57)(32,69,58)(33,70,59)(34,71,60)(35,72,49)(36,61,50), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,63)(2,62)(3,61)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44)>;

G:=Group( (1,19,42)(2,20,43)(3,21,44)(4,22,45)(5,23,46)(6,24,47)(7,13,48)(8,14,37)(9,15,38)(10,16,39)(11,17,40)(12,18,41)(25,62,51)(26,63,52)(27,64,53)(28,65,54)(29,66,55)(30,67,56)(31,68,57)(32,69,58)(33,70,59)(34,71,60)(35,72,49)(36,61,50), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,63)(2,62)(3,61)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44) );

G=PermutationGroup([[(1,19,42),(2,20,43),(3,21,44),(4,22,45),(5,23,46),(6,24,47),(7,13,48),(8,14,37),(9,15,38),(10,16,39),(11,17,40),(12,18,41),(25,62,51),(26,63,52),(27,64,53),(28,65,54),(29,66,55),(30,67,56),(31,68,57),(32,69,58),(33,70,59),(34,71,60),(35,72,49),(36,61,50)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,63),(2,62),(3,61),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,60),(24,59),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,48),(33,47),(34,46),(35,45),(36,44)]])

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 size 1 1 6 6 9 9 54 54 2 ··· 2 4 4 4 4 2 18 2 ··· 2 4 4 4 4 12 ··· 12 18 18 2 2 4 ··· 4 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D12 S32 S3×D4 C2×S32 S3×D12 kernel C3⋊S3×D12 C33⋊7D4 C32×D12 C12×C3⋊S3 C33⋊12D4 C2×S3×C3⋊S3 C3×D12 C4×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C3×C12 S3×C6 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C3 # reps 1 2 1 1 1 2 4 1 2 1 5 8 1 4 4 4 4 8

Matrix representation of C3⋊S3×D12 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 3 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 1

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

C3⋊S3×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C3:S3xD12");
// GroupNames label

G:=SmallGroup(432,672);
// by ID

G=gap.SmallGroup(432,672);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,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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