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G = S3×C12⋊S3order 432 = 24·33

Direct product of S3 and C12⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: S3×C12⋊S3, C124S32, C32(S3×D12), (S3×C12)⋊7S3, (C3×S3)⋊1D12, (C3×C12)⋊11D6, C3313(C2×D4), (S3×C6).44D6, (S3×C32)⋊5D4, C3221(S3×D4), C326(C2×D12), C338D41C2, (C3×Dic3)⋊12D6, C3312D47C2, (C32×C12)⋊3C22, (C32×C6).49C23, (C32×Dic3)⋊12C22, C41(S3×C3⋊S3), C121(C2×C3⋊S3), (S3×C3×C12)⋊8C2, C6.59(C2×S32), (C2×C3⋊S3)⋊14D6, (C4×S3)⋊3(C3⋊S3), C31(C2×C12⋊S3), (C6×C3⋊S3)⋊8C22, Dic33(C2×C3⋊S3), D6.10(C2×C3⋊S3), (C3×C12⋊S3)⋊10C2, C6.12(C22×C3⋊S3), (S3×C3×C6).28C22, (C3×C6).106(C22×S3), (C2×C33⋊C2)⋊4C22, (C2×S3×C3⋊S3)⋊5C2, C2.15(C2×S3×C3⋊S3), SmallGroup(432,671)

Series: Derived Chief Lower central Upper central

C1C32×C6 — S3×C12⋊S3
C1C3C32C33C32×C6S3×C3×C6C2×S3×C3⋊S3 — S3×C12⋊S3
C33C32×C6 — S3×C12⋊S3
C1C2C4

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 [×6], C3, C3 [×4], C3 [×4], C4, C4, C22 [×9], S3 [×2], S3 [×26], C6, C6 [×4], C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C32 [×4], Dic3, C12, C12 [×4], C12 [×8], D6, D6 [×46], C2×C6 [×6], C2×D4, C3×S3 [×8], C3×S3 [×8], C3⋊S3 [×20], C3×C6, C3×C6 [×4], C3×C6 [×6], C4×S3, D12 [×21], C3⋊D4 [×2], C2×C12 [×4], C3×D4, C22×S3 [×10], C33, C3×Dic3 [×4], C3×C12, C3×C12 [×4], C3×C12 [×5], S32 [×16], S3×C6 [×4], S3×C6 [×8], C2×C3⋊S3 [×2], C2×C3⋊S3 [×22], C62, C2×D12 [×4], S3×D4, S3×C32 [×2], C3×C3⋊S3 [×2], C33⋊C2 [×2], C32×C6, C3⋊D12 [×8], S3×C12 [×4], C3×D12 [×4], C12⋊S3, C12⋊S3 [×11], C6×C12, C2×S32 [×8], C22×C3⋊S3 [×2], C32×Dic3, C32×C12, S3×C3⋊S3 [×4], S3×C3×C6, C6×C3⋊S3 [×2], C2×C33⋊C2 [×2], S3×D12 [×4], C2×C12⋊S3, C338D4 [×2], S3×C3×C12, C3×C12⋊S3, C3312D4, C2×S3×C3⋊S3 [×2], S3×C12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], D4 [×2], C23, D6 [×15], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], C2×D12 [×4], S3×D4, C12⋊S3 [×2], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, S3×D12 [×4], C2×C12⋊S3, C2×S3×C3⋊S3, S3×C12⋊S3

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

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F6G6H6I6J···6Q6R6S12A···12H12I···12Q12R···12Y
order122222223···33333446···666666···66612···1212···1212···12
size1133181854542···24444262···244446···636362···24···46···6

63 irreducible representations

dim111111222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D6D12S32S3×D4C2×S32S3×D12
kernelS3×C12⋊S3C338D4S3×C3×C12C3×C12⋊S3C3312D4C2×S3×C3⋊S3S3×C12C12⋊S3S3×C32C3×Dic3C3×C12S3×C6C2×C3⋊S3C3×S3C12C32C6C3
# reps1211124124542164148

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

100000
010000
001000
000100
0000121
0000120
,
100000
010000
0012000
0001200
000001
000010
,
730000
10100000
001000
000100
000010
000001
,
1210000
1200000
0012100
0012000
000010
000001
,
0120000
1200000
0001200
0012000
000010
000001

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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