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## G = S3×C12⋊S3order 432 = 24·33

### Direct product of S3 and C12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — S3×C12⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — S3×C12⋊S3
 Lower central C33 — C32×C6 — S3×C12⋊S3
 Upper central C1 — C2 — C4

Generators and relations for S3×C12⋊S3
G = < a,b,c,d,e | a3=b2=c12=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2944 in 388 conjugacy classes, 80 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, 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, C3⋊D4, C2×C12, C3×D4, C22×S3, C33, 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, C12⋊S3, C12⋊S3, C6×C12, C2×S32, C22×C3⋊S3, C32×Dic3, C32×C12, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, S3×D12, C2×C12⋊S3, C338D4, S3×C3×C12, C3×C12⋊S3, C3312D4, C2×S3×C3⋊S3, S3×C12⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, S32, C2×C3⋊S3, C2×D12, S3×D4, C12⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×D12, C2×C12⋊S3, C2×S3×C3⋊S3, S3×C12⋊S3

Smallest permutation representation of S3×C12⋊S3
On 72 points
Generators in S72
(1 46 24)(2 47 13)(3 48 14)(4 37 15)(5 38 16)(6 39 17)(7 40 18)(8 41 19)(9 42 20)(10 43 21)(11 44 22)(12 45 23)(25 68 54)(26 69 55)(27 70 56)(28 71 57)(29 72 58)(30 61 59)(31 62 60)(32 63 49)(33 64 50)(34 65 51)(35 66 52)(36 67 53)
(1 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 20 38)(2 21 39)(3 22 40)(4 23 41)(5 24 42)(6 13 43)(7 14 44)(8 15 45)(9 16 46)(10 17 47)(11 18 48)(12 19 37)(25 64 58)(26 65 59)(27 66 60)(28 67 49)(29 68 50)(30 69 51)(31 70 52)(32 71 53)(33 72 54)(34 61 55)(35 62 56)(36 63 57)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 72)(47 71)(48 70)

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

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

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

63 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 ··· 12H 12I ··· 12Q 12R ··· 12Y 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 3 3 18 18 54 54 2 ··· 2 4 4 4 4 2 6 2 ··· 2 4 4 4 4 6 ··· 6 36 36 2 ··· 2 4 ··· 4 6 ··· 6

63 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 S3×C12⋊S3 C33⋊8D4 S3×C3×C12 C3×C12⋊S3 C33⋊12D4 C2×S3×C3⋊S3 S3×C12 C12⋊S3 S3×C32 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 4 5 4 2 16 4 1 4 8

Matrix representation of S3×C12⋊S3 in GL6(𝔽13)

 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 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 7 3 0 0 0 0 10 10 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 1
,
 12 1 0 0 0 0 12 0 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
,
 0 12 0 0 0 0 12 0 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))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[7,10,0,0,0,0,3,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×C12⋊S3 in GAP, Magma, Sage, TeX

S_3\times C_{12}\rtimes S_3
% in TeX

G:=Group("S3xC12:S3");
// GroupNames label

G:=SmallGroup(432,671);
// by ID

G=gap.SmallGroup(432,671);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,571,2028,14118]);
// Polycyclic

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

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