metabelian, supersoluble, monomial
Aliases: C36:1S3, C3:1D36, C12:1D9, C9:1D12, C18.16D6, C6.16D18, C32.4D12, C4:(C9:S3), (C3xC9):6D4, (C3xC36):3C2, (C3xC6).52D6, C12.2(C3:S3), C3.(C12:S3), (C3xC12).10S3, (C3xC18).20C22, (C2xC9:S3):3C2, C2.4(C2xC9:S3), C6.10(C2xC3:S3), SmallGroup(216,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36:S3
G = < a,b,c | a36=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 562 in 80 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C9, C32, C12, C12, D6, D9, C18, C3:S3, C3xC6, D12, C3xC9, C36, D18, C3xC12, C2xC3:S3, C9:S3, C3xC18, D36, C12:S3, C3xC36, C2xC9:S3, C36:S3
Quotients: C1, C2, C22, S3, D4, D6, D9, C3:S3, D12, D18, C2xC3:S3, C9:S3, D36, C12:S3, C2xC9:S3, C36:S3
►On 108 points
Generators in S108
(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 60 106)(2 61 107)(3 62 108)(4 63 73)(5 64 74)(6 65 75)(7 66 76)(8 67 77)(9 68 78)(10 69 79)(11 70 80)(12 71 81)(13 72 82)(14 37 83)(15 38 84)(16 39 85)(17 40 86)(18 41 87)(19 42 88)(20 43 89)(21 44 90)(22 45 91)(23 46 92)(24 47 93)(25 48 94)(26 49 95)(27 50 96)(28 51 97)(29 52 98)(30 53 99)(31 54 100)(32 55 101)(33 56 102)(34 57 103)(35 58 104)(36 59 105)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 96)(71 95)(72 94)
G:=sub<Sym(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), (1,60,106)(2,61,107)(3,62,108)(4,63,73)(5,64,74)(6,65,75)(7,66,76)(8,67,77)(9,68,78)(10,69,79)(11,70,80)(12,71,81)(13,72,82)(14,37,83)(15,38,84)(16,39,85)(17,40,86)(18,41,87)(19,42,88)(20,43,89)(21,44,90)(22,45,91)(23,46,92)(24,47,93)(25,48,94)(26,49,95)(27,50,96)(28,51,97)(29,52,98)(30,53,99)(31,54,100)(32,55,101)(33,56,102)(34,57,103)(35,58,104)(36,59,105), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)>;
G:=Group( (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), (1,60,106)(2,61,107)(3,62,108)(4,63,73)(5,64,74)(6,65,75)(7,66,76)(8,67,77)(9,68,78)(10,69,79)(11,70,80)(12,71,81)(13,72,82)(14,37,83)(15,38,84)(16,39,85)(17,40,86)(18,41,87)(19,42,88)(20,43,89)(21,44,90)(22,45,91)(23,46,92)(24,47,93)(25,48,94)(26,49,95)(27,50,96)(28,51,97)(29,52,98)(30,53,99)(31,54,100)(32,55,101)(33,56,102)(34,57,103)(35,58,104)(36,59,105), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94) );
G=PermutationGroup([[(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)], [(1,60,106),(2,61,107),(3,62,108),(4,63,73),(5,64,74),(6,65,75),(7,66,76),(8,67,77),(9,68,78),(10,69,79),(11,70,80),(12,71,81),(13,72,82),(14,37,83),(15,38,84),(16,39,85),(17,40,86),(18,41,87),(19,42,88),(20,43,89),(21,44,90),(22,45,91),(23,46,92),(24,47,93),(25,48,94),(26,49,95),(27,50,96),(28,51,97),(29,52,98),(30,53,99),(31,54,100),(32,55,101),(33,56,102),(34,57,103),(35,58,104),(36,59,105)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,108),(59,107),(60,106),(61,105),(62,104),(63,103),(64,102),(65,101),(66,100),(67,99),(68,98),(69,97),(70,96),(71,95),(72,94)]])
C36:S3 is a maximal subgroup of
C6.D36 C3:D72 C9:D24 C18.D12 C24:D9 C72:1S3 C36.18D6 C36.20D6 Dic6:5D9 Dic9.D6 S3xD36 D9xD12 C36.70D6 D4xC9:S3 C36.29D6
C36:S3 is a maximal quotient of
C24.D9 C24:D9 C72:1S3 C36:Dic3 C6.11D36
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D9 | D12 | D12 | D18 | D36 |
| kernel | C36:S3 | C3xC36 | C2xC9:S3 | C36 | C3xC12 | C3xC9 | C18 | C3xC6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 9 | 6 | 2 | 9 | 18 |
Matrix representation of C36:S3 ►in GL4(F37) generated by
| 6 | 17 | 0 | 0 |
| 20 | 26 | 0 | 0 |
| 0 | 0 | 4 | 25 |
| 0 | 0 | 12 | 29 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 1 | 36 |
| 11 | 17 | 0 | 0 |
| 6 | 26 | 0 | 0 |
| 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,4,12,0,0,25,29],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,36,36],[11,6,0,0,17,26,0,0,0,0,36,0,0,0,1,1] >;
C36:S3 in GAP, Magma, Sage, TeX
C_{36}\rtimes S_3 % in TeX
G:=Group("C36:S3"); // GroupNames label
G:=SmallGroup(216,65);
// by ID
G=gap.SmallGroup(216,65);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,2115,453,1444,5189]);
// Polycyclic
G:=Group<a,b,c|a^36=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations