direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C12×Dic9, C36⋊6C12, C62.118D6, C9⋊3(C4×C12), (C3×C36)⋊4C4, (C3×C9)⋊3C42, C2.2(C12×D9), C6.23(C4×D9), C6.11(S3×C12), (C6×C12).47S3, (C2×C36).20C6, (C6×C36).14C2, (C2×C12).22D9, (C2×C6).46D18, C6.8(C6×Dic3), C2.2(C6×Dic9), C22.3(C6×D9), C18.17(C2×C12), (C6×Dic9).9C2, (C2×Dic9).8C6, C6.19(C2×Dic9), C3.1(Dic3×C12), (C6×C18).32C22, C12.12(C3×Dic3), (C3×C12).24Dic3, C32.3(C4×Dic3), (C2×C4).6(C3×D9), (C2×C6).34(S3×C6), (C3×C6).65(C4×S3), (C2×C12).30(C3×S3), (C3×C18).20(C2×C4), (C2×C18).23(C2×C6), (C3×C6).56(C2×Dic3), SmallGroup(432,128)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C12×Dic9 |
Generators and relations for C12×Dic9
G = < a,b,c | a12=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 262 in 106 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C3×C18, C3×C18, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C3×Dic9, C3×C36, C6×C18, C4×Dic9, Dic3×C12, C6×Dic9, C6×C36, C12×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, D9, C3×S3, C4×S3, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C3×D9, C4×D9, C2×Dic9, S3×C12, C6×Dic3, C3×Dic9, C6×D9, C4×Dic9, Dic3×C12, C12×D9, C6×Dic9, C12×Dic9
►On 144 points
Generators in S144
(1 39 76 140 13 51 88 134 7 45 82 128)(2 40 77 141 14 52 89 135 8 46 83 129)(3 41 78 142 15 53 90 136 9 47 84 130)(4 42 79 143 16 54 73 137 10 48 85 131)(5 43 80 144 17 37 74 138 11 49 86 132)(6 44 81 127 18 38 75 139 12 50 87 133)(19 112 106 68 25 118 94 56 31 124 100 62)(20 113 107 69 26 119 95 57 32 125 101 63)(21 114 108 70 27 120 96 58 33 126 102 64)(22 115 91 71 28 121 97 59 34 109 103 65)(23 116 92 72 29 122 98 60 35 110 104 66)(24 117 93 55 30 123 99 61 36 111 105 67)
(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 26 10 35)(2 25 11 34)(3 24 12 33)(4 23 13 32)(5 22 14 31)(6 21 15 30)(7 20 16 29)(8 19 17 28)(9 36 18 27)(37 121 46 112)(38 120 47 111)(39 119 48 110)(40 118 49 109)(41 117 50 126)(42 116 51 125)(43 115 52 124)(44 114 53 123)(45 113 54 122)(55 133 64 142)(56 132 65 141)(57 131 66 140)(58 130 67 139)(59 129 68 138)(60 128 69 137)(61 127 70 136)(62 144 71 135)(63 143 72 134)(73 98 82 107)(74 97 83 106)(75 96 84 105)(76 95 85 104)(77 94 86 103)(78 93 87 102)(79 92 88 101)(80 91 89 100)(81 108 90 99)
G:=sub<Sym(144)| (1,39,76,140,13,51,88,134,7,45,82,128)(2,40,77,141,14,52,89,135,8,46,83,129)(3,41,78,142,15,53,90,136,9,47,84,130)(4,42,79,143,16,54,73,137,10,48,85,131)(5,43,80,144,17,37,74,138,11,49,86,132)(6,44,81,127,18,38,75,139,12,50,87,133)(19,112,106,68,25,118,94,56,31,124,100,62)(20,113,107,69,26,119,95,57,32,125,101,63)(21,114,108,70,27,120,96,58,33,126,102,64)(22,115,91,71,28,121,97,59,34,109,103,65)(23,116,92,72,29,122,98,60,35,110,104,66)(24,117,93,55,30,123,99,61,36,111,105,67), (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), (1,26,10,35)(2,25,11,34)(3,24,12,33)(4,23,13,32)(5,22,14,31)(6,21,15,30)(7,20,16,29)(8,19,17,28)(9,36,18,27)(37,121,46,112)(38,120,47,111)(39,119,48,110)(40,118,49,109)(41,117,50,126)(42,116,51,125)(43,115,52,124)(44,114,53,123)(45,113,54,122)(55,133,64,142)(56,132,65,141)(57,131,66,140)(58,130,67,139)(59,129,68,138)(60,128,69,137)(61,127,70,136)(62,144,71,135)(63,143,72,134)(73,98,82,107)(74,97,83,106)(75,96,84,105)(76,95,85,104)(77,94,86,103)(78,93,87,102)(79,92,88,101)(80,91,89,100)(81,108,90,99)>;
G:=Group( (1,39,76,140,13,51,88,134,7,45,82,128)(2,40,77,141,14,52,89,135,8,46,83,129)(3,41,78,142,15,53,90,136,9,47,84,130)(4,42,79,143,16,54,73,137,10,48,85,131)(5,43,80,144,17,37,74,138,11,49,86,132)(6,44,81,127,18,38,75,139,12,50,87,133)(19,112,106,68,25,118,94,56,31,124,100,62)(20,113,107,69,26,119,95,57,32,125,101,63)(21,114,108,70,27,120,96,58,33,126,102,64)(22,115,91,71,28,121,97,59,34,109,103,65)(23,116,92,72,29,122,98,60,35,110,104,66)(24,117,93,55,30,123,99,61,36,111,105,67), (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), (1,26,10,35)(2,25,11,34)(3,24,12,33)(4,23,13,32)(5,22,14,31)(6,21,15,30)(7,20,16,29)(8,19,17,28)(9,36,18,27)(37,121,46,112)(38,120,47,111)(39,119,48,110)(40,118,49,109)(41,117,50,126)(42,116,51,125)(43,115,52,124)(44,114,53,123)(45,113,54,122)(55,133,64,142)(56,132,65,141)(57,131,66,140)(58,130,67,139)(59,129,68,138)(60,128,69,137)(61,127,70,136)(62,144,71,135)(63,143,72,134)(73,98,82,107)(74,97,83,106)(75,96,84,105)(76,95,85,104)(77,94,86,103)(78,93,87,102)(79,92,88,101)(80,91,89,100)(81,108,90,99) );
G=PermutationGroup([[(1,39,76,140,13,51,88,134,7,45,82,128),(2,40,77,141,14,52,89,135,8,46,83,129),(3,41,78,142,15,53,90,136,9,47,84,130),(4,42,79,143,16,54,73,137,10,48,85,131),(5,43,80,144,17,37,74,138,11,49,86,132),(6,44,81,127,18,38,75,139,12,50,87,133),(19,112,106,68,25,118,94,56,31,124,100,62),(20,113,107,69,26,119,95,57,32,125,101,63),(21,114,108,70,27,120,96,58,33,126,102,64),(22,115,91,71,28,121,97,59,34,109,103,65),(23,116,92,72,29,122,98,60,35,110,104,66),(24,117,93,55,30,123,99,61,36,111,105,67)], [(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)], [(1,26,10,35),(2,25,11,34),(3,24,12,33),(4,23,13,32),(5,22,14,31),(6,21,15,30),(7,20,16,29),(8,19,17,28),(9,36,18,27),(37,121,46,112),(38,120,47,111),(39,119,48,110),(40,118,49,109),(41,117,50,126),(42,116,51,125),(43,115,52,124),(44,114,53,123),(45,113,54,122),(55,133,64,142),(56,132,65,141),(57,131,66,140),(58,130,67,139),(59,129,68,138),(60,128,69,137),(61,127,70,136),(62,144,71,135),(63,143,72,134),(73,98,82,107),(74,97,83,106),(75,96,84,105),(76,95,85,104),(77,94,86,103),(78,93,87,102),(79,92,88,101),(80,91,89,100),(81,108,90,99)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 9A | ··· | 9I | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AJ | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | D9 | C3×S3 | C4×S3 | Dic9 | C3×Dic3 | D18 | S3×C6 | C3×D9 | C4×D9 | S3×C12 | C3×Dic9 | C6×D9 | C12×D9 |
| kernel | C12×Dic9 | C6×Dic9 | C6×C36 | C4×Dic9 | C3×Dic9 | C3×C36 | C2×Dic9 | C2×C36 | Dic9 | C36 | C6×C12 | C3×C12 | C62 | C2×C12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 2 | 1 | 3 | 2 | 4 | 6 | 4 | 3 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C12×Dic9 ►in GL4(𝔽37) generated by
| 8 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [8,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,36,0,0,0,0,16,0,0,0,0,7],[1,0,0,0,0,31,0,0,0,0,0,36,0,0,36,0] >;
C12×Dic9 in GAP, Magma, Sage, TeX
C_{12}\times {\rm Dic}_9 % in TeX
G:=Group("C12xDic9"); // GroupNames label
G:=SmallGroup(432,128);
// by ID
G=gap.SmallGroup(432,128);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^12=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations