direct product, metabelian, supersoluble, monomial, A-group
Aliases: C6×C9⋊S3, C32⋊8D18, C33.12D6, C6⋊(C3×D9), C9⋊5(S3×C6), C3⋊2(C6×D9), (C3×C6)⋊3D9, C18⋊3(C3×S3), (C3×C9)⋊19D6, (C3×C18)⋊15C6, (C3×C18)⋊10S3, (C32×C18)⋊4C2, (C32×C9)⋊9C22, C32.19(S3×C6), (C32×C6).19S3, C6.5(C3×C3⋊S3), C3.1(C6×C3⋊S3), (C3×C9)⋊17(C2×C6), (C3×C6).38(C3×S3), (C3×C6).21(C3⋊S3), C32.11(C2×C3⋊S3), SmallGroup(324,142)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C6×C9⋊S3 |
Generators and relations for C6×C9⋊S3
G = < a,b,c,d | a6=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 556 in 130 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, C32, D6, C2×C6, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, C33, D18, S3×C6, C2×C3⋊S3, C3×D9, C9⋊S3, C3×C18, C3×C18, C3×C18, C3×C3⋊S3, C32×C6, C32×C9, C6×D9, C2×C9⋊S3, C6×C3⋊S3, C3×C9⋊S3, C32×C18, C6×C9⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, D9, C3×S3, C3⋊S3, D18, S3×C6, C2×C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C6×D9, C2×C9⋊S3, C6×C3⋊S3, C3×C9⋊S3, C6×C9⋊S3
►On 108 points
Generators in S108
(1 62 28 67 43 48)(2 63 29 68 44 49)(3 55 30 69 45 50)(4 56 31 70 37 51)(5 57 32 71 38 52)(6 58 33 72 39 53)(7 59 34 64 40 54)(8 60 35 65 41 46)(9 61 36 66 42 47)(10 78 24 99 105 86)(11 79 25 91 106 87)(12 80 26 92 107 88)(13 81 27 93 108 89)(14 73 19 94 100 90)(15 74 20 95 101 82)(16 75 21 96 102 83)(17 76 22 97 103 84)(18 77 23 98 104 85)
(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 34 37)(2 35 38)(3 36 39)(4 28 40)(5 29 41)(6 30 42)(7 31 43)(8 32 44)(9 33 45)(10 102 27)(11 103 19)(12 104 20)(13 105 21)(14 106 22)(15 107 23)(16 108 24)(17 100 25)(18 101 26)(46 57 68)(47 58 69)(48 59 70)(49 60 71)(50 61 72)(51 62 64)(52 63 65)(53 55 66)(54 56 67)(73 87 97)(74 88 98)(75 89 99)(76 90 91)(77 82 92)(78 83 93)(79 84 94)(80 85 95)(81 86 96)
(1 74)(2 73)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 54)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 95)(29 94)(30 93)(31 92)(32 91)(33 99)(34 98)(35 97)(36 96)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 90)(45 89)(64 104)(65 103)(66 102)(67 101)(68 100)(69 108)(70 107)(71 106)(72 105)
G:=sub<Sym(108)| (1,62,28,67,43,48)(2,63,29,68,44,49)(3,55,30,69,45,50)(4,56,31,70,37,51)(5,57,32,71,38,52)(6,58,33,72,39,53)(7,59,34,64,40,54)(8,60,35,65,41,46)(9,61,36,66,42,47)(10,78,24,99,105,86)(11,79,25,91,106,87)(12,80,26,92,107,88)(13,81,27,93,108,89)(14,73,19,94,100,90)(15,74,20,95,101,82)(16,75,21,96,102,83)(17,76,22,97,103,84)(18,77,23,98,104,85), (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), (1,34,37)(2,35,38)(3,36,39)(4,28,40)(5,29,41)(6,30,42)(7,31,43)(8,32,44)(9,33,45)(10,102,27)(11,103,19)(12,104,20)(13,105,21)(14,106,22)(15,107,23)(16,108,24)(17,100,25)(18,101,26)(46,57,68)(47,58,69)(48,59,70)(49,60,71)(50,61,72)(51,62,64)(52,63,65)(53,55,66)(54,56,67)(73,87,97)(74,88,98)(75,89,99)(76,90,91)(77,82,92)(78,83,93)(79,84,94)(80,85,95)(81,86,96), (1,74)(2,73)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,54)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,95)(29,94)(30,93)(31,92)(32,91)(33,99)(34,98)(35,97)(36,96)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,90)(45,89)(64,104)(65,103)(66,102)(67,101)(68,100)(69,108)(70,107)(71,106)(72,105)>;
G:=Group( (1,62,28,67,43,48)(2,63,29,68,44,49)(3,55,30,69,45,50)(4,56,31,70,37,51)(5,57,32,71,38,52)(6,58,33,72,39,53)(7,59,34,64,40,54)(8,60,35,65,41,46)(9,61,36,66,42,47)(10,78,24,99,105,86)(11,79,25,91,106,87)(12,80,26,92,107,88)(13,81,27,93,108,89)(14,73,19,94,100,90)(15,74,20,95,101,82)(16,75,21,96,102,83)(17,76,22,97,103,84)(18,77,23,98,104,85), (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), (1,34,37)(2,35,38)(3,36,39)(4,28,40)(5,29,41)(6,30,42)(7,31,43)(8,32,44)(9,33,45)(10,102,27)(11,103,19)(12,104,20)(13,105,21)(14,106,22)(15,107,23)(16,108,24)(17,100,25)(18,101,26)(46,57,68)(47,58,69)(48,59,70)(49,60,71)(50,61,72)(51,62,64)(52,63,65)(53,55,66)(54,56,67)(73,87,97)(74,88,98)(75,89,99)(76,90,91)(77,82,92)(78,83,93)(79,84,94)(80,85,95)(81,86,96), (1,74)(2,73)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,54)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,95)(29,94)(30,93)(31,92)(32,91)(33,99)(34,98)(35,97)(36,96)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,90)(45,89)(64,104)(65,103)(66,102)(67,101)(68,100)(69,108)(70,107)(71,106)(72,105) );
G=PermutationGroup([[(1,62,28,67,43,48),(2,63,29,68,44,49),(3,55,30,69,45,50),(4,56,31,70,37,51),(5,57,32,71,38,52),(6,58,33,72,39,53),(7,59,34,64,40,54),(8,60,35,65,41,46),(9,61,36,66,42,47),(10,78,24,99,105,86),(11,79,25,91,106,87),(12,80,26,92,107,88),(13,81,27,93,108,89),(14,73,19,94,100,90),(15,74,20,95,101,82),(16,75,21,96,102,83),(17,76,22,97,103,84),(18,77,23,98,104,85)], [(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)], [(1,34,37),(2,35,38),(3,36,39),(4,28,40),(5,29,41),(6,30,42),(7,31,43),(8,32,44),(9,33,45),(10,102,27),(11,103,19),(12,104,20),(13,105,21),(14,106,22),(15,107,23),(16,108,24),(17,100,25),(18,101,26),(46,57,68),(47,58,69),(48,59,70),(49,60,71),(50,61,72),(51,62,64),(52,63,65),(53,55,66),(54,56,67),(73,87,97),(74,88,98),(75,89,99),(76,90,91),(77,82,92),(78,83,93),(79,84,94),(80,85,95),(81,86,96)], [(1,74),(2,73),(3,81),(4,80),(5,79),(6,78),(7,77),(8,76),(9,75),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,54),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,95),(29,94),(30,93),(31,92),(32,91),(33,99),(34,98),(35,97),(36,96),(37,88),(38,87),(39,86),(40,85),(41,84),(42,83),(43,82),(44,90),(45,89),(64,104),(65,103),(66,102),(67,101),(68,100),(69,108),(70,107),(71,106),(72,105)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 6Q | 6R | 9A | ··· | 9AA | 18A | ··· | 18AA |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 27 | 27 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | S3 | D6 | D6 | C3×S3 | D9 | C3×S3 | S3×C6 | D18 | S3×C6 | C3×D9 | C6×D9 |
| kernel | C6×C9⋊S3 | C3×C9⋊S3 | C32×C18 | C2×C9⋊S3 | C9⋊S3 | C3×C18 | C3×C18 | C32×C6 | C3×C9 | C33 | C18 | C3×C6 | C3×C6 | C9 | C32 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 3 | 1 | 3 | 1 | 6 | 9 | 2 | 6 | 9 | 2 | 18 | 18 |
Matrix representation of C6×C9⋊S3 ►in GL5(𝔽19)
| 8 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(19))| [8,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,6,0,0,0,0,0,16,0,0,0,0,0,6,0,0,0,0,0,16],[1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C6×C9⋊S3 in GAP, Magma, Sage, TeX
C_6\times C_9\rtimes S_3
% in TeX
G:=Group("C6xC9:S3"); // GroupNames label
G:=SmallGroup(324,142);
// by ID
G=gap.SmallGroup(324,142);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,453,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations