metabelian, supersoluble, monomial
Aliases: C36⋊1S3, C3⋊1D36, C12⋊1D9, C9⋊1D12, C18.16D6, C6.16D18, C32.4D12, C4⋊(C9⋊S3), (C3×C9)⋊6D4, (C3×C36)⋊3C2, (C3×C6).52D6, C12.2(C3⋊S3), C3.(C12⋊S3), (C3×C12).10S3, (C3×C18).20C22, (C2×C9⋊S3)⋊3C2, C2.4(C2×C9⋊S3), C6.10(C2×C3⋊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 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×8], C6, C6 [×3], D4, C9 [×3], C32, C12, C12 [×3], D6 [×8], D9 [×6], C18 [×3], C3⋊S3 [×2], C3×C6, D12 [×4], C3×C9, C36 [×3], D18 [×6], C3×C12, C2×C3⋊S3 [×2], C9⋊S3 [×2], C3×C18, D36 [×3], C12⋊S3, C3×C36, C2×C9⋊S3 [×2], C36⋊S3
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D9 [×3], C3⋊S3, D12 [×4], D18 [×3], C2×C3⋊S3, C9⋊S3, D36 [×3], C12⋊S3, C2×C9⋊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 90 67)(2 91 68)(3 92 69)(4 93 70)(5 94 71)(6 95 72)(7 96 37)(8 97 38)(9 98 39)(10 99 40)(11 100 41)(12 101 42)(13 102 43)(14 103 44)(15 104 45)(16 105 46)(17 106 47)(18 107 48)(19 108 49)(20 73 50)(21 74 51)(22 75 52)(23 76 53)(24 77 54)(25 78 55)(26 79 56)(27 80 57)(28 81 58)(29 82 59)(30 83 60)(31 84 61)(32 85 62)(33 86 63)(34 87 64)(35 88 65)(36 89 66)
(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 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)
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), (1,90,67)(2,91,68)(3,92,69)(4,93,70)(5,94,71)(6,95,72)(7,96,37)(8,97,38)(9,98,39)(10,99,40)(11,100,41)(12,101,42)(13,102,43)(14,103,44)(15,104,45)(16,105,46)(17,106,47)(18,107,48)(19,108,49)(20,73,50)(21,74,51)(22,75,52)(23,76,53)(24,77,54)(25,78,55)(26,79,56)(27,80,57)(28,81,58)(29,82,59)(30,83,60)(31,84,61)(32,85,62)(33,86,63)(34,87,64)(35,88,65)(36,89,66), (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)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)>;
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), (1,90,67)(2,91,68)(3,92,69)(4,93,70)(5,94,71)(6,95,72)(7,96,37)(8,97,38)(9,98,39)(10,99,40)(11,100,41)(12,101,42)(13,102,43)(14,103,44)(15,104,45)(16,105,46)(17,106,47)(18,107,48)(19,108,49)(20,73,50)(21,74,51)(22,75,52)(23,76,53)(24,77,54)(25,78,55)(26,79,56)(27,80,57)(28,81,58)(29,82,59)(30,83,60)(31,84,61)(32,85,62)(33,86,63)(34,87,64)(35,88,65)(36,89,66), (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)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85) );
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)], [(1,90,67),(2,91,68),(3,92,69),(4,93,70),(5,94,71),(6,95,72),(7,96,37),(8,97,38),(9,98,39),(10,99,40),(11,100,41),(12,101,42),(13,102,43),(14,103,44),(15,104,45),(16,105,46),(17,106,47),(18,107,48),(19,108,49),(20,73,50),(21,74,51),(22,75,52),(23,76,53),(24,77,54),(25,78,55),(26,79,56),(27,80,57),(28,81,58),(29,82,59),(30,83,60),(31,84,61),(32,85,62),(33,86,63),(34,87,64),(35,88,65),(36,89,66)], [(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),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,108),(50,107),(51,106),(52,105),(53,104),(54,103),(55,102),(56,101),(57,100),(58,99),(59,98),(60,97),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85)])
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 S3×D36 D9×D12 C36.70D6 D4×C9⋊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 | C3×C36 | C2×C9⋊S3 | C36 | C3×C12 | C3×C9 | C18 | C3×C6 | 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