metabelian, supersoluble, monomial
Aliases: C3⋊2Dic36, C6.15D36, C36.33D6, C12.12D18, Dic18.1S3, C32.2Dic12, C3⋊C8.D9, (C3×C9)⋊1Q16, C12.47S32, (C3×C18).4D4, C4.11(S3×D9), C9⋊1(C3⋊Q16), (C3×C6).32D12, (C3×C12).72D6, C18.4(C3⋊D4), (C3×C36).4C22, C6.4(C3⋊D12), C2.7(C3⋊D36), C12.D9.1C2, (C3×Dic18).1C2, C3.1(C32⋊3Q16), (C9×C3⋊C8).1C2, (C3×C3⋊C8).2S3, SmallGroup(432,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Dic36
G = < a,b,c | a3=b72=1, c2=b36, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 400 in 66 conjugacy classes, 25 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×C9, Dic9, C36, C36, C3×Dic3, C3⋊Dic3, C3×C12, Dic12, C3⋊Q16, C3×C18, C72, Dic18, Dic18, C3×C3⋊C8, C3×Dic6, C32⋊4Q8, C3×Dic9, C9⋊Dic3, C3×C36, Dic36, C32⋊3Q16, C9×C3⋊C8, C3×Dic18, C12.D9, C3⋊Dic36
Quotients: C1, C2, C22, S3, D4, D6, Q16, D9, D12, C3⋊D4, D18, S32, Dic12, C3⋊Q16, D36, C3⋊D12, S3×D9, Dic36, C32⋊3Q16, C3⋊D36, C3⋊Dic36
►On 144 points
Generators in S144
(1 49 25)(2 26 50)(3 51 27)(4 28 52)(5 53 29)(6 30 54)(7 55 31)(8 32 56)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(17 65 41)(18 42 66)(19 67 43)(20 44 68)(21 69 45)(22 46 70)(23 71 47)(24 48 72)(73 97 121)(74 122 98)(75 99 123)(76 124 100)(77 101 125)(78 126 102)(79 103 127)(80 128 104)(81 105 129)(82 130 106)(83 107 131)(84 132 108)(85 109 133)(86 134 110)(87 111 135)(88 136 112)(89 113 137)(90 138 114)(91 115 139)(92 140 116)(93 117 141)(94 142 118)(95 119 143)(96 144 120)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 99 37 135)(2 98 38 134)(3 97 39 133)(4 96 40 132)(5 95 41 131)(6 94 42 130)(7 93 43 129)(8 92 44 128)(9 91 45 127)(10 90 46 126)(11 89 47 125)(12 88 48 124)(13 87 49 123)(14 86 50 122)(15 85 51 121)(16 84 52 120)(17 83 53 119)(18 82 54 118)(19 81 55 117)(20 80 56 116)(21 79 57 115)(22 78 58 114)(23 77 59 113)(24 76 60 112)(25 75 61 111)(26 74 62 110)(27 73 63 109)(28 144 64 108)(29 143 65 107)(30 142 66 106)(31 141 67 105)(32 140 68 104)(33 139 69 103)(34 138 70 102)(35 137 71 101)(36 136 72 100)
G:=sub<Sym(144)| (1,49,25)(2,26,50)(3,51,27)(4,28,52)(5,53,29)(6,30,54)(7,55,31)(8,32,56)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,65,41)(18,42,66)(19,67,43)(20,44,68)(21,69,45)(22,46,70)(23,71,47)(24,48,72)(73,97,121)(74,122,98)(75,99,123)(76,124,100)(77,101,125)(78,126,102)(79,103,127)(80,128,104)(81,105,129)(82,130,106)(83,107,131)(84,132,108)(85,109,133)(86,134,110)(87,111,135)(88,136,112)(89,113,137)(90,138,114)(91,115,139)(92,140,116)(93,117,141)(94,142,118)(95,119,143)(96,144,120), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,99,37,135)(2,98,38,134)(3,97,39,133)(4,96,40,132)(5,95,41,131)(6,94,42,130)(7,93,43,129)(8,92,44,128)(9,91,45,127)(10,90,46,126)(11,89,47,125)(12,88,48,124)(13,87,49,123)(14,86,50,122)(15,85,51,121)(16,84,52,120)(17,83,53,119)(18,82,54,118)(19,81,55,117)(20,80,56,116)(21,79,57,115)(22,78,58,114)(23,77,59,113)(24,76,60,112)(25,75,61,111)(26,74,62,110)(27,73,63,109)(28,144,64,108)(29,143,65,107)(30,142,66,106)(31,141,67,105)(32,140,68,104)(33,139,69,103)(34,138,70,102)(35,137,71,101)(36,136,72,100)>;
G:=Group( (1,49,25)(2,26,50)(3,51,27)(4,28,52)(5,53,29)(6,30,54)(7,55,31)(8,32,56)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,65,41)(18,42,66)(19,67,43)(20,44,68)(21,69,45)(22,46,70)(23,71,47)(24,48,72)(73,97,121)(74,122,98)(75,99,123)(76,124,100)(77,101,125)(78,126,102)(79,103,127)(80,128,104)(81,105,129)(82,130,106)(83,107,131)(84,132,108)(85,109,133)(86,134,110)(87,111,135)(88,136,112)(89,113,137)(90,138,114)(91,115,139)(92,140,116)(93,117,141)(94,142,118)(95,119,143)(96,144,120), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,99,37,135)(2,98,38,134)(3,97,39,133)(4,96,40,132)(5,95,41,131)(6,94,42,130)(7,93,43,129)(8,92,44,128)(9,91,45,127)(10,90,46,126)(11,89,47,125)(12,88,48,124)(13,87,49,123)(14,86,50,122)(15,85,51,121)(16,84,52,120)(17,83,53,119)(18,82,54,118)(19,81,55,117)(20,80,56,116)(21,79,57,115)(22,78,58,114)(23,77,59,113)(24,76,60,112)(25,75,61,111)(26,74,62,110)(27,73,63,109)(28,144,64,108)(29,143,65,107)(30,142,66,106)(31,141,67,105)(32,140,68,104)(33,139,69,103)(34,138,70,102)(35,137,71,101)(36,136,72,100) );
G=PermutationGroup([[(1,49,25),(2,26,50),(3,51,27),(4,28,52),(5,53,29),(6,30,54),(7,55,31),(8,32,56),(9,57,33),(10,34,58),(11,59,35),(12,36,60),(13,61,37),(14,38,62),(15,63,39),(16,40,64),(17,65,41),(18,42,66),(19,67,43),(20,44,68),(21,69,45),(22,46,70),(23,71,47),(24,48,72),(73,97,121),(74,122,98),(75,99,123),(76,124,100),(77,101,125),(78,126,102),(79,103,127),(80,128,104),(81,105,129),(82,130,106),(83,107,131),(84,132,108),(85,109,133),(86,134,110),(87,111,135),(88,136,112),(89,113,137),(90,138,114),(91,115,139),(92,140,116),(93,117,141),(94,142,118),(95,119,143),(96,144,120)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,99,37,135),(2,98,38,134),(3,97,39,133),(4,96,40,132),(5,95,41,131),(6,94,42,130),(7,93,43,129),(8,92,44,128),(9,91,45,127),(10,90,46,126),(11,89,47,125),(12,88,48,124),(13,87,49,123),(14,86,50,122),(15,85,51,121),(16,84,52,120),(17,83,53,119),(18,82,54,118),(19,81,55,117),(20,80,56,116),(21,79,57,115),(22,78,58,114),(23,77,59,113),(24,76,60,112),(25,75,61,111),(26,74,62,110),(27,73,63,109),(28,144,64,108),(29,143,65,107),(30,142,66,106),(31,141,67,105),(32,140,68,104),(33,139,69,103),(34,138,70,102),(35,137,71,101),(36,136,72,100)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 4 | 2 | 36 | 108 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | Q16 | D9 | C3⋊D4 | D12 | D18 | Dic12 | D36 | Dic36 | S32 | C3⋊Q16 | C3⋊D12 | S3×D9 | C32⋊3Q16 | C3⋊D36 | C3⋊Dic36 |
| kernel | C3⋊Dic36 | C9×C3⋊C8 | C3×Dic18 | C12.D9 | Dic18 | C3×C3⋊C8 | C3×C18 | C36 | C3×C12 | C3×C9 | C3⋊C8 | C18 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 4 | 6 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of C3⋊Dic36 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 50 | 18 | 0 | 0 | 0 | 0 |
| 55 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 19 |
| 0 | 0 | 0 | 0 | 54 | 48 |
| 18 | 53 | 0 | 0 | 0 | 0 |
| 71 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,55,0,0,0,0,18,68,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,29,54,0,0,0,0,19,48],[18,71,0,0,0,0,53,55,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;
C3⋊Dic36 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_{36} % in TeX
G:=Group("C3:Dic36"); // GroupNames label
G:=SmallGroup(432,65);
// by ID
G=gap.SmallGroup(432,65);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^3=b^72=1,c^2=b^36,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations