metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic12⋊9C4, C8.9(C4×S3), (C2×C8).58D6, C6.50(C4×D4), C4.Q8.2S3, C24.24(C2×C4), C4⋊C4.159D6, C24⋊C4.1C2, C3⋊3(Q16⋊C4), Dic6.8(C2×C4), C22.82(S3×D4), C12.26(C4○D4), C12.41(C22×C4), C4.2(Q8⋊3S3), C6.SD16.4C2, C2.5(D4.D6), (C2×C12).269C23, (C2×C24).107C22, (C2×Dic12).13C2, (C2×Dic3).160D4, Dic6⋊C4.4C2, C6.37(C8.C22), C2.10(Dic3⋊5D4), (C4×Dic3).27C22, (C2×Dic6).79C22, C4.41(S3×C2×C4), (C3×C4.Q8).2C2, (C2×C6).274(C2×D4), (C2×C3⋊C8).51C22, (C3×C4⋊C4).62C22, (C2×C4).372(C22×S3), SmallGroup(192,412)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic12⋊9C4
G = < a,b,c | a24=c4=1, b2=a12, bab-1=a-1, cac-1=a19, cbc-1=a12b >
Subgroups: 256 in 108 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, Q16⋊C4, C6.SD16, C24⋊C4, C3×C4.Q8, Dic6⋊C4, C2×Dic12, Dic12⋊9C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8.C22, S3×C2×C4, S3×D4, Q8⋊3S3, Q16⋊C4, Dic3⋊5D4, D4.D6, Dic12⋊9C4
►Regular action on 192 points
Generators in S192
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
(1 66 13 54)(2 65 14 53)(3 64 15 52)(4 63 16 51)(5 62 17 50)(6 61 18 49)(7 60 19 72)(8 59 20 71)(9 58 21 70)(10 57 22 69)(11 56 23 68)(12 55 24 67)(25 117 37 105)(26 116 38 104)(27 115 39 103)(28 114 40 102)(29 113 41 101)(30 112 42 100)(31 111 43 99)(32 110 44 98)(33 109 45 97)(34 108 46 120)(35 107 47 119)(36 106 48 118)(73 130 85 142)(74 129 86 141)(75 128 87 140)(76 127 88 139)(77 126 89 138)(78 125 90 137)(79 124 91 136)(80 123 92 135)(81 122 93 134)(82 121 94 133)(83 144 95 132)(84 143 96 131)(145 186 157 174)(146 185 158 173)(147 184 159 172)(148 183 160 171)(149 182 161 170)(150 181 162 169)(151 180 163 192)(152 179 164 191)(153 178 165 190)(154 177 166 189)(155 176 167 188)(156 175 168 187)
(1 120 187 144)(2 115 188 139)(3 110 189 134)(4 105 190 129)(5 100 191 124)(6 119 192 143)(7 114 169 138)(8 109 170 133)(9 104 171 128)(10 99 172 123)(11 118 173 142)(12 113 174 137)(13 108 175 132)(14 103 176 127)(15 98 177 122)(16 117 178 141)(17 112 179 136)(18 107 180 131)(19 102 181 126)(20 97 182 121)(21 116 183 140)(22 111 184 135)(23 106 185 130)(24 101 186 125)(25 165 86 51)(26 160 87 70)(27 155 88 65)(28 150 89 60)(29 145 90 55)(30 164 91 50)(31 159 92 69)(32 154 93 64)(33 149 94 59)(34 168 95 54)(35 163 96 49)(36 158 73 68)(37 153 74 63)(38 148 75 58)(39 167 76 53)(40 162 77 72)(41 157 78 67)(42 152 79 62)(43 147 80 57)(44 166 81 52)(45 161 82 71)(46 156 83 66)(47 151 84 61)(48 146 85 56)
G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,66,13,54)(2,65,14,53)(3,64,15,52)(4,63,16,51)(5,62,17,50)(6,61,18,49)(7,60,19,72)(8,59,20,71)(9,58,21,70)(10,57,22,69)(11,56,23,68)(12,55,24,67)(25,117,37,105)(26,116,38,104)(27,115,39,103)(28,114,40,102)(29,113,41,101)(30,112,42,100)(31,111,43,99)(32,110,44,98)(33,109,45,97)(34,108,46,120)(35,107,47,119)(36,106,48,118)(73,130,85,142)(74,129,86,141)(75,128,87,140)(76,127,88,139)(77,126,89,138)(78,125,90,137)(79,124,91,136)(80,123,92,135)(81,122,93,134)(82,121,94,133)(83,144,95,132)(84,143,96,131)(145,186,157,174)(146,185,158,173)(147,184,159,172)(148,183,160,171)(149,182,161,170)(150,181,162,169)(151,180,163,192)(152,179,164,191)(153,178,165,190)(154,177,166,189)(155,176,167,188)(156,175,168,187), (1,120,187,144)(2,115,188,139)(3,110,189,134)(4,105,190,129)(5,100,191,124)(6,119,192,143)(7,114,169,138)(8,109,170,133)(9,104,171,128)(10,99,172,123)(11,118,173,142)(12,113,174,137)(13,108,175,132)(14,103,176,127)(15,98,177,122)(16,117,178,141)(17,112,179,136)(18,107,180,131)(19,102,181,126)(20,97,182,121)(21,116,183,140)(22,111,184,135)(23,106,185,130)(24,101,186,125)(25,165,86,51)(26,160,87,70)(27,155,88,65)(28,150,89,60)(29,145,90,55)(30,164,91,50)(31,159,92,69)(32,154,93,64)(33,149,94,59)(34,168,95,54)(35,163,96,49)(36,158,73,68)(37,153,74,63)(38,148,75,58)(39,167,76,53)(40,162,77,72)(41,157,78,67)(42,152,79,62)(43,147,80,57)(44,166,81,52)(45,161,82,71)(46,156,83,66)(47,151,84,61)(48,146,85,56)>;
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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,66,13,54)(2,65,14,53)(3,64,15,52)(4,63,16,51)(5,62,17,50)(6,61,18,49)(7,60,19,72)(8,59,20,71)(9,58,21,70)(10,57,22,69)(11,56,23,68)(12,55,24,67)(25,117,37,105)(26,116,38,104)(27,115,39,103)(28,114,40,102)(29,113,41,101)(30,112,42,100)(31,111,43,99)(32,110,44,98)(33,109,45,97)(34,108,46,120)(35,107,47,119)(36,106,48,118)(73,130,85,142)(74,129,86,141)(75,128,87,140)(76,127,88,139)(77,126,89,138)(78,125,90,137)(79,124,91,136)(80,123,92,135)(81,122,93,134)(82,121,94,133)(83,144,95,132)(84,143,96,131)(145,186,157,174)(146,185,158,173)(147,184,159,172)(148,183,160,171)(149,182,161,170)(150,181,162,169)(151,180,163,192)(152,179,164,191)(153,178,165,190)(154,177,166,189)(155,176,167,188)(156,175,168,187), (1,120,187,144)(2,115,188,139)(3,110,189,134)(4,105,190,129)(5,100,191,124)(6,119,192,143)(7,114,169,138)(8,109,170,133)(9,104,171,128)(10,99,172,123)(11,118,173,142)(12,113,174,137)(13,108,175,132)(14,103,176,127)(15,98,177,122)(16,117,178,141)(17,112,179,136)(18,107,180,131)(19,102,181,126)(20,97,182,121)(21,116,183,140)(22,111,184,135)(23,106,185,130)(24,101,186,125)(25,165,86,51)(26,160,87,70)(27,155,88,65)(28,150,89,60)(29,145,90,55)(30,164,91,50)(31,159,92,69)(32,154,93,64)(33,149,94,59)(34,168,95,54)(35,163,96,49)(36,158,73,68)(37,153,74,63)(38,148,75,58)(39,167,76,53)(40,162,77,72)(41,157,78,67)(42,152,79,62)(43,147,80,57)(44,166,81,52)(45,161,82,71)(46,156,83,66)(47,151,84,61)(48,146,85,56) );
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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)], [(1,66,13,54),(2,65,14,53),(3,64,15,52),(4,63,16,51),(5,62,17,50),(6,61,18,49),(7,60,19,72),(8,59,20,71),(9,58,21,70),(10,57,22,69),(11,56,23,68),(12,55,24,67),(25,117,37,105),(26,116,38,104),(27,115,39,103),(28,114,40,102),(29,113,41,101),(30,112,42,100),(31,111,43,99),(32,110,44,98),(33,109,45,97),(34,108,46,120),(35,107,47,119),(36,106,48,118),(73,130,85,142),(74,129,86,141),(75,128,87,140),(76,127,88,139),(77,126,89,138),(78,125,90,137),(79,124,91,136),(80,123,92,135),(81,122,93,134),(82,121,94,133),(83,144,95,132),(84,143,96,131),(145,186,157,174),(146,185,158,173),(147,184,159,172),(148,183,160,171),(149,182,161,170),(150,181,162,169),(151,180,163,192),(152,179,164,191),(153,178,165,190),(154,177,166,189),(155,176,167,188),(156,175,168,187)], [(1,120,187,144),(2,115,188,139),(3,110,189,134),(4,105,190,129),(5,100,191,124),(6,119,192,143),(7,114,169,138),(8,109,170,133),(9,104,171,128),(10,99,172,123),(11,118,173,142),(12,113,174,137),(13,108,175,132),(14,103,176,127),(15,98,177,122),(16,117,178,141),(17,112,179,136),(18,107,180,131),(19,102,181,126),(20,97,182,121),(21,116,183,140),(22,111,184,135),(23,106,185,130),(24,101,186,125),(25,165,86,51),(26,160,87,70),(27,155,88,65),(28,150,89,60),(29,145,90,55),(30,164,91,50),(31,159,92,69),(32,154,93,64),(33,149,94,59),(34,168,95,54),(35,163,96,49),(36,158,73,68),(37,153,74,63),(38,148,75,58),(39,167,76,53),(40,162,77,72),(41,157,78,67),(42,152,79,62),(43,147,80,57),(44,166,81,52),(45,161,82,71),(46,156,83,66),(47,151,84,61),(48,146,85,56)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C8.C22 | Q8⋊3S3 | S3×D4 | D4.D6 |
| kernel | Dic12⋊9C4 | C6.SD16 | C24⋊C4 | C3×C4.Q8 | Dic6⋊C4 | C2×Dic12 | Dic12 | C4.Q8 | C2×Dic3 | C4⋊C4 | C2×C8 | C12 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 4 |
Matrix representation of Dic12⋊9C4 ►in GL8(𝔽73)
| 5 | 54 | 38 | 42 | 0 | 0 | 0 | 0 |
| 19 | 24 | 31 | 69 | 0 | 0 | 0 | 0 |
| 38 | 42 | 68 | 19 | 0 | 0 | 0 | 0 |
| 31 | 69 | 54 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 18 | 58 | 72 |
| 0 | 0 | 0 | 0 | 64 | 0 | 21 | 15 |
| 0 | 0 | 0 | 0 | 57 | 31 | 0 | 18 |
| 0 | 0 | 0 | 0 | 36 | 16 | 64 | 18 |
| 22 | 21 | 36 | 17 | 0 | 0 | 0 | 0 |
| 72 | 51 | 54 | 37 | 0 | 0 | 0 | 0 |
| 37 | 56 | 22 | 21 | 0 | 0 | 0 | 0 |
| 19 | 36 | 72 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 69 | 20 | 70 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 | 0 | 70 |
| 0 | 0 | 0 | 0 | 70 | 0 | 4 | 53 |
| 0 | 0 | 0 | 0 | 0 | 70 | 67 | 69 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
G:=sub<GL(8,GF(73))| [5,19,38,31,0,0,0,0,54,24,42,69,0,0,0,0,38,31,68,54,0,0,0,0,42,69,19,49,0,0,0,0,0,0,0,0,55,64,57,36,0,0,0,0,18,0,31,16,0,0,0,0,58,21,0,64,0,0,0,0,72,15,18,18],[22,72,37,19,0,0,0,0,21,51,56,36,0,0,0,0,36,54,22,72,0,0,0,0,17,37,21,51,0,0,0,0,0,0,0,0,69,6,70,0,0,0,0,0,20,4,0,70,0,0,0,0,70,0,4,67,0,0,0,0,0,70,53,69],[0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
Dic12⋊9C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{12}\rtimes_9C_4 % in TeX
G:=Group("Dic12:9C4"); // GroupNames label
G:=SmallGroup(192,412);
// by ID
G=gap.SmallGroup(192,412);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,758,135,100,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=c^4=1,b^2=a^12,b*a*b^-1=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^12*b>;
// generators/relations