direct product, metabelian, supersoluble, monomial, A-group
Aliases: C22×C9⋊S3, C18⋊2D6, C6⋊2D18, C62.14S3, (C2×C6)⋊5D9, (C2×C18)⋊5S3, (C6×C18)⋊7C2, (C3×C9)⋊4C23, C9⋊2(C22×S3), (C3×C6).56D6, C3⋊2(C22×D9), (C3×C18)⋊4C22, C32.4(C22×S3), C3.(C22×C3⋊S3), C6.13(C2×C3⋊S3), (C2×C6).7(C3⋊S3), SmallGroup(216,112)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C22×C9⋊S3 |
Generators and relations for C22×C9⋊S3
G = < a,b,c,d,e | a2=b2=c9=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 886 in 160 conjugacy classes, 61 normal (9 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C23, C9, C32, D6, C2×C6, C2×C6, D9, C18, C3⋊S3, C3×C6, C22×S3, C3×C9, D18, C2×C18, C2×C3⋊S3, C62, C9⋊S3, C3×C18, C22×D9, C22×C3⋊S3, C2×C9⋊S3, C6×C18, C22×C9⋊S3
Quotients: C1, C2, C22, S3, C23, D6, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, C22×C9⋊S3
►On 108 points
Generators in S108
(1 89)(2 90)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 59)(11 60)(12 61)(13 62)(14 63)(15 55)(16 56)(17 57)(18 58)(19 71)(20 72)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 73)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 62)(2 63)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 86)(11 87)(12 88)(13 89)(14 90)(15 82)(16 83)(17 84)(18 85)(19 98)(20 99)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 46)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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 30 37)(2 31 38)(3 32 39)(4 33 40)(5 34 41)(6 35 42)(7 36 43)(8 28 44)(9 29 45)(10 100 27)(11 101 19)(12 102 20)(13 103 21)(14 104 22)(15 105 23)(16 106 24)(17 107 25)(18 108 26)(46 70 59)(47 71 60)(48 72 61)(49 64 62)(50 65 63)(51 66 55)(52 67 56)(53 68 57)(54 69 58)(73 97 86)(74 98 87)(75 99 88)(76 91 89)(77 92 90)(78 93 82)(79 94 83)(80 95 84)(81 96 85)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 108)(26 107)(27 106)(28 39)(29 38)(30 37)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(46 67)(47 66)(48 65)(49 64)(50 72)(51 71)(52 70)(53 69)(54 68)(55 60)(56 59)(57 58)(61 63)(73 94)(74 93)(75 92)(76 91)(77 99)(78 98)(79 97)(80 96)(81 95)(82 87)(83 86)(84 85)(88 90)
G:=sub<Sym(108)| (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,59)(11,60)(12,61)(13,62)(14,63)(15,55)(16,56)(17,57)(18,58)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,62)(2,63)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,86)(11,87)(12,88)(13,89)(14,90)(15,82)(16,83)(17,84)(18,85)(19,98)(20,99)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,46)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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), (1,30,37)(2,31,38)(3,32,39)(4,33,40)(5,34,41)(6,35,42)(7,36,43)(8,28,44)(9,29,45)(10,100,27)(11,101,19)(12,102,20)(13,103,21)(14,104,22)(15,105,23)(16,106,24)(17,107,25)(18,108,26)(46,70,59)(47,71,60)(48,72,61)(49,64,62)(50,65,63)(51,66,55)(52,67,56)(53,68,57)(54,69,58)(73,97,86)(74,98,87)(75,99,88)(76,91,89)(77,92,90)(78,93,82)(79,94,83)(80,95,84)(81,96,85), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,108)(26,107)(27,106)(28,39)(29,38)(30,37)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(46,67)(47,66)(48,65)(49,64)(50,72)(51,71)(52,70)(53,69)(54,68)(55,60)(56,59)(57,58)(61,63)(73,94)(74,93)(75,92)(76,91)(77,99)(78,98)(79,97)(80,96)(81,95)(82,87)(83,86)(84,85)(88,90)>;
G:=Group( (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,59)(11,60)(12,61)(13,62)(14,63)(15,55)(16,56)(17,57)(18,58)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,62)(2,63)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,86)(11,87)(12,88)(13,89)(14,90)(15,82)(16,83)(17,84)(18,85)(19,98)(20,99)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,46)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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), (1,30,37)(2,31,38)(3,32,39)(4,33,40)(5,34,41)(6,35,42)(7,36,43)(8,28,44)(9,29,45)(10,100,27)(11,101,19)(12,102,20)(13,103,21)(14,104,22)(15,105,23)(16,106,24)(17,107,25)(18,108,26)(46,70,59)(47,71,60)(48,72,61)(49,64,62)(50,65,63)(51,66,55)(52,67,56)(53,68,57)(54,69,58)(73,97,86)(74,98,87)(75,99,88)(76,91,89)(77,92,90)(78,93,82)(79,94,83)(80,95,84)(81,96,85), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,108)(26,107)(27,106)(28,39)(29,38)(30,37)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(46,67)(47,66)(48,65)(49,64)(50,72)(51,71)(52,70)(53,69)(54,68)(55,60)(56,59)(57,58)(61,63)(73,94)(74,93)(75,92)(76,91)(77,99)(78,98)(79,97)(80,96)(81,95)(82,87)(83,86)(84,85)(88,90) );
G=PermutationGroup([[(1,89),(2,90),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,59),(11,60),(12,61),(13,62),(14,63),(15,55),(16,56),(17,57),(18,58),(19,71),(20,72),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,73),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(1,62),(2,63),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,86),(11,87),(12,88),(13,89),(14,90),(15,82),(16,83),(17,84),(18,85),(19,98),(20,99),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,46),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108)], [(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)], [(1,30,37),(2,31,38),(3,32,39),(4,33,40),(5,34,41),(6,35,42),(7,36,43),(8,28,44),(9,29,45),(10,100,27),(11,101,19),(12,102,20),(13,103,21),(14,104,22),(15,105,23),(16,106,24),(17,107,25),(18,108,26),(46,70,59),(47,71,60),(48,72,61),(49,64,62),(50,65,63),(51,66,55),(52,67,56),(53,68,57),(54,69,58),(73,97,86),(74,98,87),(75,99,88),(76,91,89),(77,92,90),(78,93,82),(79,94,83),(80,95,84),(81,96,85)], [(2,9),(3,8),(4,7),(5,6),(10,16),(11,15),(12,14),(17,18),(19,105),(20,104),(21,103),(22,102),(23,101),(24,100),(25,108),(26,107),(27,106),(28,39),(29,38),(30,37),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(46,67),(47,66),(48,65),(49,64),(50,72),(51,71),(52,70),(53,69),(54,68),(55,60),(56,59),(57,58),(61,63),(73,94),(74,93),(75,92),(76,91),(77,99),(78,98),(79,97),(80,96),(81,95),(82,87),(83,86),(84,85),(88,90)]])
C22×C9⋊S3 is a maximal subgroup of
C6.18D36 C6.11D36 D18⋊D6 C22×S3×D9
C22×C9⋊S3 is a maximal quotient of C36.70D6 C36.27D6 C36.29D6
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 6A | ··· | 6L | 9A | ··· | 9I | 18A | ··· | 18AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | S3 | D6 | D6 | D9 | D18 |
| kernel | C22×C9⋊S3 | C2×C9⋊S3 | C6×C18 | C2×C18 | C62 | C18 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 6 | 1 | 3 | 1 | 9 | 3 | 9 | 27 |
Matrix representation of C22×C9⋊S3 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 17 | 0 | 0 |
| 0 | 2 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 18 | 17 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 17 | 16 |
| 0 | 0 | 0 | 1 | 1 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 14 | 12 | 0 | 0 |
| 0 | 17 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 18 | 18 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,12,2,0,0,0,17,14,0,0,0,0,0,1,18,0,0,0,3,17],[1,0,0,0,0,0,0,18,0,0,0,1,18,0,0,0,0,0,17,1,0,0,0,16,1],[18,0,0,0,0,0,14,17,0,0,0,12,5,0,0,0,0,0,1,18,0,0,0,0,18] >;
C22×C9⋊S3 in GAP, Magma, Sage, TeX
C_2^2\times C_9\rtimes S_3
% in TeX
G:=Group("C2^2xC9:S3"); // GroupNames label
G:=SmallGroup(216,112);
// by ID
G=gap.SmallGroup(216,112);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,2115,453,1444,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^9=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,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations