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G = S3×C32×C6order 324 = 22·34

Direct product of C32×C6 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×C32×C6, C347C22, C324C62, C6⋊(C32×C6), C3⋊(C3×C62), (C33×C6)⋊1C2, (C32×C6)⋊7C6, C3311(C2×C6), (C3×C6)⋊3(C3×C6), SmallGroup(324,172)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C32×C6
C1C3C32C33C34S3×C33 — S3×C32×C6
C3 — S3×C32×C6
C1C32×C6

Generators and relations for S3×C32×C6
 G = < a,b,c,d,e | a3=b3=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 760 in 436 conjugacy classes, 196 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×13], C3 [×13], C22, S3 [×2], C6, C6 [×13], C6 [×39], C32 [×26], C32 [×52], D6, C2×C6 [×13], C3×S3 [×26], C3×C6 [×26], C3×C6 [×78], C33, C33 [×13], C33 [×13], S3×C6 [×13], C62 [×13], S3×C32 [×26], C32×C6, C32×C6 [×13], C32×C6 [×15], C34, S3×C3×C6 [×13], C3×C62, S3×C33 [×2], C33×C6, S3×C32×C6
Quotients: C1, C2 [×3], C3 [×13], C22, S3, C6 [×39], C32 [×13], D6, C2×C6 [×13], C3×S3 [×13], C3×C6 [×39], C33, S3×C6 [×13], C62 [×13], S3×C32 [×13], C32×C6 [×3], S3×C3×C6 [×13], C3×C62, S3×C33, S3×C32×C6

Smallest permutation representation of S3×C32×C6
On 108 points
Generators in S108
(1 23 58)(2 24 59)(3 19 60)(4 20 55)(5 21 56)(6 22 57)(7 106 94)(8 107 95)(9 108 96)(10 103 91)(11 104 92)(12 105 93)(13 90 52)(14 85 53)(15 86 54)(16 87 49)(17 88 50)(18 89 51)(25 66 73)(26 61 74)(27 62 75)(28 63 76)(29 64 77)(30 65 78)(31 79 98)(32 80 99)(33 81 100)(34 82 101)(35 83 102)(36 84 97)(37 71 46)(38 72 47)(39 67 48)(40 68 43)(41 69 44)(42 70 45)
(1 40 51)(2 41 52)(3 42 53)(4 37 54)(5 38 49)(6 39 50)(7 30 83)(8 25 84)(9 26 79)(10 27 80)(11 28 81)(12 29 82)(13 24 69)(14 19 70)(15 20 71)(16 21 72)(17 22 67)(18 23 68)(31 96 74)(32 91 75)(33 92 76)(34 93 77)(35 94 78)(36 95 73)(43 89 58)(44 90 59)(45 85 60)(46 86 55)(47 87 56)(48 88 57)(61 98 108)(62 99 103)(63 100 104)(64 101 105)(65 102 106)(66 97 107)
(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)
(1 42 49)(2 37 50)(3 38 51)(4 39 52)(5 40 53)(6 41 54)(7 81 26)(8 82 27)(9 83 28)(10 84 29)(11 79 30)(12 80 25)(13 20 67)(14 21 68)(15 22 69)(16 23 70)(17 24 71)(18 19 72)(31 78 92)(32 73 93)(33 74 94)(34 75 95)(35 76 96)(36 77 91)(43 85 56)(44 86 57)(45 87 58)(46 88 59)(47 89 60)(48 90 55)(61 106 100)(62 107 101)(63 108 102)(64 103 97)(65 104 98)(66 105 99)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 76)(20 77)(21 78)(22 73)(23 74)(24 75)(25 57)(26 58)(27 59)(28 60)(29 55)(30 56)(31 68)(32 69)(33 70)(34 71)(35 72)(36 67)(37 101)(38 102)(39 97)(40 98)(41 99)(42 100)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 106)(50 107)(51 108)(52 103)(53 104)(54 105)

G:=sub<Sym(108)| (1,23,58)(2,24,59)(3,19,60)(4,20,55)(5,21,56)(6,22,57)(7,106,94)(8,107,95)(9,108,96)(10,103,91)(11,104,92)(12,105,93)(13,90,52)(14,85,53)(15,86,54)(16,87,49)(17,88,50)(18,89,51)(25,66,73)(26,61,74)(27,62,75)(28,63,76)(29,64,77)(30,65,78)(31,79,98)(32,80,99)(33,81,100)(34,82,101)(35,83,102)(36,84,97)(37,71,46)(38,72,47)(39,67,48)(40,68,43)(41,69,44)(42,70,45), (1,40,51)(2,41,52)(3,42,53)(4,37,54)(5,38,49)(6,39,50)(7,30,83)(8,25,84)(9,26,79)(10,27,80)(11,28,81)(12,29,82)(13,24,69)(14,19,70)(15,20,71)(16,21,72)(17,22,67)(18,23,68)(31,96,74)(32,91,75)(33,92,76)(34,93,77)(35,94,78)(36,95,73)(43,89,58)(44,90,59)(45,85,60)(46,86,55)(47,87,56)(48,88,57)(61,98,108)(62,99,103)(63,100,104)(64,101,105)(65,102,106)(66,97,107), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,42,49)(2,37,50)(3,38,51)(4,39,52)(5,40,53)(6,41,54)(7,81,26)(8,82,27)(9,83,28)(10,84,29)(11,79,30)(12,80,25)(13,20,67)(14,21,68)(15,22,69)(16,23,70)(17,24,71)(18,19,72)(31,78,92)(32,73,93)(33,74,94)(34,75,95)(35,76,96)(36,77,91)(43,85,56)(44,86,57)(45,87,58)(46,88,59)(47,89,60)(48,90,55)(61,106,100)(62,107,101)(63,108,102)(64,103,97)(65,104,98)(66,105,99), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,76)(20,77)(21,78)(22,73)(23,74)(24,75)(25,57)(26,58)(27,59)(28,60)(29,55)(30,56)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(37,101)(38,102)(39,97)(40,98)(41,99)(42,100)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,106)(50,107)(51,108)(52,103)(53,104)(54,105)>;

G:=Group( (1,23,58)(2,24,59)(3,19,60)(4,20,55)(5,21,56)(6,22,57)(7,106,94)(8,107,95)(9,108,96)(10,103,91)(11,104,92)(12,105,93)(13,90,52)(14,85,53)(15,86,54)(16,87,49)(17,88,50)(18,89,51)(25,66,73)(26,61,74)(27,62,75)(28,63,76)(29,64,77)(30,65,78)(31,79,98)(32,80,99)(33,81,100)(34,82,101)(35,83,102)(36,84,97)(37,71,46)(38,72,47)(39,67,48)(40,68,43)(41,69,44)(42,70,45), (1,40,51)(2,41,52)(3,42,53)(4,37,54)(5,38,49)(6,39,50)(7,30,83)(8,25,84)(9,26,79)(10,27,80)(11,28,81)(12,29,82)(13,24,69)(14,19,70)(15,20,71)(16,21,72)(17,22,67)(18,23,68)(31,96,74)(32,91,75)(33,92,76)(34,93,77)(35,94,78)(36,95,73)(43,89,58)(44,90,59)(45,85,60)(46,86,55)(47,87,56)(48,88,57)(61,98,108)(62,99,103)(63,100,104)(64,101,105)(65,102,106)(66,97,107), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,42,49)(2,37,50)(3,38,51)(4,39,52)(5,40,53)(6,41,54)(7,81,26)(8,82,27)(9,83,28)(10,84,29)(11,79,30)(12,80,25)(13,20,67)(14,21,68)(15,22,69)(16,23,70)(17,24,71)(18,19,72)(31,78,92)(32,73,93)(33,74,94)(34,75,95)(35,76,96)(36,77,91)(43,85,56)(44,86,57)(45,87,58)(46,88,59)(47,89,60)(48,90,55)(61,106,100)(62,107,101)(63,108,102)(64,103,97)(65,104,98)(66,105,99), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,76)(20,77)(21,78)(22,73)(23,74)(24,75)(25,57)(26,58)(27,59)(28,60)(29,55)(30,56)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(37,101)(38,102)(39,97)(40,98)(41,99)(42,100)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,106)(50,107)(51,108)(52,103)(53,104)(54,105) );

G=PermutationGroup([(1,23,58),(2,24,59),(3,19,60),(4,20,55),(5,21,56),(6,22,57),(7,106,94),(8,107,95),(9,108,96),(10,103,91),(11,104,92),(12,105,93),(13,90,52),(14,85,53),(15,86,54),(16,87,49),(17,88,50),(18,89,51),(25,66,73),(26,61,74),(27,62,75),(28,63,76),(29,64,77),(30,65,78),(31,79,98),(32,80,99),(33,81,100),(34,82,101),(35,83,102),(36,84,97),(37,71,46),(38,72,47),(39,67,48),(40,68,43),(41,69,44),(42,70,45)], [(1,40,51),(2,41,52),(3,42,53),(4,37,54),(5,38,49),(6,39,50),(7,30,83),(8,25,84),(9,26,79),(10,27,80),(11,28,81),(12,29,82),(13,24,69),(14,19,70),(15,20,71),(16,21,72),(17,22,67),(18,23,68),(31,96,74),(32,91,75),(33,92,76),(34,93,77),(35,94,78),(36,95,73),(43,89,58),(44,90,59),(45,85,60),(46,86,55),(47,87,56),(48,88,57),(61,98,108),(62,99,103),(63,100,104),(64,101,105),(65,102,106),(66,97,107)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108)], [(1,42,49),(2,37,50),(3,38,51),(4,39,52),(5,40,53),(6,41,54),(7,81,26),(8,82,27),(9,83,28),(10,84,29),(11,79,30),(12,80,25),(13,20,67),(14,21,68),(15,22,69),(16,23,70),(17,24,71),(18,19,72),(31,78,92),(32,73,93),(33,74,94),(34,75,95),(35,76,96),(36,77,91),(43,85,56),(44,86,57),(45,87,58),(46,88,59),(47,89,60),(48,90,55),(61,106,100),(62,107,101),(63,108,102),(64,103,97),(65,104,98),(66,105,99)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,87),(8,88),(9,89),(10,90),(11,85),(12,86),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,76),(20,77),(21,78),(22,73),(23,74),(24,75),(25,57),(26,58),(27,59),(28,60),(29,55),(30,56),(31,68),(32,69),(33,70),(34,71),(35,72),(36,67),(37,101),(38,102),(39,97),(40,98),(41,99),(42,100),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,106),(50,107),(51,108),(52,103),(53,104),(54,105)])

162 conjugacy classes

class 1 2A2B2C3A···3Z3AA···3BA6A···6Z6AA···6BA6BB···6DA
order12223···33···36···66···66···6
size11331···12···21···12···23···3

162 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6
kernelS3×C32×C6S3×C33C33×C6S3×C3×C6S3×C32C32×C6C32×C6C33C3×C6C32
# reps121265226112626

Matrix representation of S3×C32×C6 in GL4(𝔽7) generated by

1000
0400
0010
0001
,
4000
0400
0010
0001
,
1000
0500
0040
0004
,
1000
0100
0020
0064
,
1000
0100
0023
0065
G:=sub<GL(4,GF(7))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,5,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,2,6,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,2,6,0,0,3,5] >;

S3×C32×C6 in GAP, Magma, Sage, TeX

S_3\times C_3^2\times C_6
% in TeX

G:=Group("S3xC3^2xC6");
// GroupNames label

G:=SmallGroup(324,172);
// by ID

G=gap.SmallGroup(324,172);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^6=d^3=e^2=1,a*b=b*a,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,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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