metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.5Dic6, C42.55D6, C3⋊6(Q8.Q8), (C3×Q8).3Q8, C4⋊C4.250D6, (C2×C12).65D4, (Q8×C12).4C2, (C4×Q8).11S3, C12.30(C2×Q8), C6.91(C4○D8), (C2×Q8).181D6, C12⋊C8.16C2, C4.14(C2×Dic6), C4.64(C4○D12), C12.57(C4○D4), (C4×C12).93C22, Q8⋊2Dic3.9C2, C6.Q16.11C2, C6.66(C22⋊Q8), C12.6Q8.6C2, (C2×C12).344C23, C6.86(C8.C22), C12.Q8.11C2, (C6×Q8).192C22, C2.12(Q8.13D6), C2.8(Q8.11D6), C4⋊Dic3.141C22, C2.17(C12.48D4), (C2×C6).475(C2×D4), (C2×C3⋊C8).99C22, (C2×C4).220(C3⋊D4), (C3×C4⋊C4).281C22, (C2×C4).444(C22×S3), C22.154(C2×C3⋊D4), SmallGroup(192,581)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.5Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=a2c6, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 200 in 90 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, Q8.Q8, C12⋊C8, C6.Q16, C12.Q8, Q8⋊2Dic3, C12.6Q8, Q8×C12, Q8.5Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C4○D8, C8.C22, C2×Dic6, C4○D12, C2×C3⋊D4, Q8.Q8, C12.48D4, Q8.11D6, Q8.13D6, Q8.5Dic6
►Regular action on 192 points
Generators in S192
(1 78 70 85)(2 79 71 86)(3 80 72 87)(4 81 61 88)(5 82 62 89)(6 83 63 90)(7 84 64 91)(8 73 65 92)(9 74 66 93)(10 75 67 94)(11 76 68 95)(12 77 69 96)(13 120 58 43)(14 109 59 44)(15 110 60 45)(16 111 49 46)(17 112 50 47)(18 113 51 48)(19 114 52 37)(20 115 53 38)(21 116 54 39)(22 117 55 40)(23 118 56 41)(24 119 57 42)(25 161 106 152)(26 162 107 153)(27 163 108 154)(28 164 97 155)(29 165 98 156)(30 166 99 145)(31 167 100 146)(32 168 101 147)(33 157 102 148)(34 158 103 149)(35 159 104 150)(36 160 105 151)(121 175 187 136)(122 176 188 137)(123 177 189 138)(124 178 190 139)(125 179 191 140)(126 180 192 141)(127 169 181 142)(128 170 182 143)(129 171 183 144)(130 172 184 133)(131 173 185 134)(132 174 186 135)
(1 109 70 44)(2 110 71 45)(3 111 72 46)(4 112 61 47)(5 113 62 48)(6 114 63 37)(7 115 64 38)(8 116 65 39)(9 117 66 40)(10 118 67 41)(11 119 68 42)(12 120 69 43)(13 96 58 77)(14 85 59 78)(15 86 60 79)(16 87 49 80)(17 88 50 81)(18 89 51 82)(19 90 52 83)(20 91 53 84)(21 92 54 73)(22 93 55 74)(23 94 56 75)(24 95 57 76)(25 138 106 177)(26 139 107 178)(27 140 108 179)(28 141 97 180)(29 142 98 169)(30 143 99 170)(31 144 100 171)(32 133 101 172)(33 134 102 173)(34 135 103 174)(35 136 104 175)(36 137 105 176)(121 159 187 150)(122 160 188 151)(123 161 189 152)(124 162 190 153)(125 163 191 154)(126 164 192 155)(127 165 181 156)(128 166 182 145)(129 167 183 146)(130 168 184 147)(131 157 185 148)(132 158 186 149)
(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)
(1 176 64 143)(2 136 65 169)(3 174 66 141)(4 134 67 179)(5 172 68 139)(6 144 69 177)(7 170 70 137)(8 142 71 175)(9 180 72 135)(10 140 61 173)(11 178 62 133)(12 138 63 171)(13 25 52 100)(14 105 53 30)(15 35 54 98)(16 103 55 28)(17 33 56 108)(18 101 57 26)(19 31 58 106)(20 99 59 36)(21 29 60 104)(22 97 49 34)(23 27 50 102)(24 107 51 32)(37 167 120 152)(38 145 109 160)(39 165 110 150)(40 155 111 158)(41 163 112 148)(42 153 113 168)(43 161 114 146)(44 151 115 166)(45 159 116 156)(46 149 117 164)(47 157 118 154)(48 147 119 162)(73 181 86 121)(74 126 87 186)(75 191 88 131)(76 124 89 184)(77 189 90 129)(78 122 91 182)(79 187 92 127)(80 132 93 192)(81 185 94 125)(82 130 95 190)(83 183 96 123)(84 128 85 188)
G:=sub<Sym(192)| (1,78,70,85)(2,79,71,86)(3,80,72,87)(4,81,61,88)(5,82,62,89)(6,83,63,90)(7,84,64,91)(8,73,65,92)(9,74,66,93)(10,75,67,94)(11,76,68,95)(12,77,69,96)(13,120,58,43)(14,109,59,44)(15,110,60,45)(16,111,49,46)(17,112,50,47)(18,113,51,48)(19,114,52,37)(20,115,53,38)(21,116,54,39)(22,117,55,40)(23,118,56,41)(24,119,57,42)(25,161,106,152)(26,162,107,153)(27,163,108,154)(28,164,97,155)(29,165,98,156)(30,166,99,145)(31,167,100,146)(32,168,101,147)(33,157,102,148)(34,158,103,149)(35,159,104,150)(36,160,105,151)(121,175,187,136)(122,176,188,137)(123,177,189,138)(124,178,190,139)(125,179,191,140)(126,180,192,141)(127,169,181,142)(128,170,182,143)(129,171,183,144)(130,172,184,133)(131,173,185,134)(132,174,186,135), (1,109,70,44)(2,110,71,45)(3,111,72,46)(4,112,61,47)(5,113,62,48)(6,114,63,37)(7,115,64,38)(8,116,65,39)(9,117,66,40)(10,118,67,41)(11,119,68,42)(12,120,69,43)(13,96,58,77)(14,85,59,78)(15,86,60,79)(16,87,49,80)(17,88,50,81)(18,89,51,82)(19,90,52,83)(20,91,53,84)(21,92,54,73)(22,93,55,74)(23,94,56,75)(24,95,57,76)(25,138,106,177)(26,139,107,178)(27,140,108,179)(28,141,97,180)(29,142,98,169)(30,143,99,170)(31,144,100,171)(32,133,101,172)(33,134,102,173)(34,135,103,174)(35,136,104,175)(36,137,105,176)(121,159,187,150)(122,160,188,151)(123,161,189,152)(124,162,190,153)(125,163,191,154)(126,164,192,155)(127,165,181,156)(128,166,182,145)(129,167,183,146)(130,168,184,147)(131,157,185,148)(132,158,186,149), (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), (1,176,64,143)(2,136,65,169)(3,174,66,141)(4,134,67,179)(5,172,68,139)(6,144,69,177)(7,170,70,137)(8,142,71,175)(9,180,72,135)(10,140,61,173)(11,178,62,133)(12,138,63,171)(13,25,52,100)(14,105,53,30)(15,35,54,98)(16,103,55,28)(17,33,56,108)(18,101,57,26)(19,31,58,106)(20,99,59,36)(21,29,60,104)(22,97,49,34)(23,27,50,102)(24,107,51,32)(37,167,120,152)(38,145,109,160)(39,165,110,150)(40,155,111,158)(41,163,112,148)(42,153,113,168)(43,161,114,146)(44,151,115,166)(45,159,116,156)(46,149,117,164)(47,157,118,154)(48,147,119,162)(73,181,86,121)(74,126,87,186)(75,191,88,131)(76,124,89,184)(77,189,90,129)(78,122,91,182)(79,187,92,127)(80,132,93,192)(81,185,94,125)(82,130,95,190)(83,183,96,123)(84,128,85,188)>;
G:=Group( (1,78,70,85)(2,79,71,86)(3,80,72,87)(4,81,61,88)(5,82,62,89)(6,83,63,90)(7,84,64,91)(8,73,65,92)(9,74,66,93)(10,75,67,94)(11,76,68,95)(12,77,69,96)(13,120,58,43)(14,109,59,44)(15,110,60,45)(16,111,49,46)(17,112,50,47)(18,113,51,48)(19,114,52,37)(20,115,53,38)(21,116,54,39)(22,117,55,40)(23,118,56,41)(24,119,57,42)(25,161,106,152)(26,162,107,153)(27,163,108,154)(28,164,97,155)(29,165,98,156)(30,166,99,145)(31,167,100,146)(32,168,101,147)(33,157,102,148)(34,158,103,149)(35,159,104,150)(36,160,105,151)(121,175,187,136)(122,176,188,137)(123,177,189,138)(124,178,190,139)(125,179,191,140)(126,180,192,141)(127,169,181,142)(128,170,182,143)(129,171,183,144)(130,172,184,133)(131,173,185,134)(132,174,186,135), (1,109,70,44)(2,110,71,45)(3,111,72,46)(4,112,61,47)(5,113,62,48)(6,114,63,37)(7,115,64,38)(8,116,65,39)(9,117,66,40)(10,118,67,41)(11,119,68,42)(12,120,69,43)(13,96,58,77)(14,85,59,78)(15,86,60,79)(16,87,49,80)(17,88,50,81)(18,89,51,82)(19,90,52,83)(20,91,53,84)(21,92,54,73)(22,93,55,74)(23,94,56,75)(24,95,57,76)(25,138,106,177)(26,139,107,178)(27,140,108,179)(28,141,97,180)(29,142,98,169)(30,143,99,170)(31,144,100,171)(32,133,101,172)(33,134,102,173)(34,135,103,174)(35,136,104,175)(36,137,105,176)(121,159,187,150)(122,160,188,151)(123,161,189,152)(124,162,190,153)(125,163,191,154)(126,164,192,155)(127,165,181,156)(128,166,182,145)(129,167,183,146)(130,168,184,147)(131,157,185,148)(132,158,186,149), (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), (1,176,64,143)(2,136,65,169)(3,174,66,141)(4,134,67,179)(5,172,68,139)(6,144,69,177)(7,170,70,137)(8,142,71,175)(9,180,72,135)(10,140,61,173)(11,178,62,133)(12,138,63,171)(13,25,52,100)(14,105,53,30)(15,35,54,98)(16,103,55,28)(17,33,56,108)(18,101,57,26)(19,31,58,106)(20,99,59,36)(21,29,60,104)(22,97,49,34)(23,27,50,102)(24,107,51,32)(37,167,120,152)(38,145,109,160)(39,165,110,150)(40,155,111,158)(41,163,112,148)(42,153,113,168)(43,161,114,146)(44,151,115,166)(45,159,116,156)(46,149,117,164)(47,157,118,154)(48,147,119,162)(73,181,86,121)(74,126,87,186)(75,191,88,131)(76,124,89,184)(77,189,90,129)(78,122,91,182)(79,187,92,127)(80,132,93,192)(81,185,94,125)(82,130,95,190)(83,183,96,123)(84,128,85,188) );
G=PermutationGroup([[(1,78,70,85),(2,79,71,86),(3,80,72,87),(4,81,61,88),(5,82,62,89),(6,83,63,90),(7,84,64,91),(8,73,65,92),(9,74,66,93),(10,75,67,94),(11,76,68,95),(12,77,69,96),(13,120,58,43),(14,109,59,44),(15,110,60,45),(16,111,49,46),(17,112,50,47),(18,113,51,48),(19,114,52,37),(20,115,53,38),(21,116,54,39),(22,117,55,40),(23,118,56,41),(24,119,57,42),(25,161,106,152),(26,162,107,153),(27,163,108,154),(28,164,97,155),(29,165,98,156),(30,166,99,145),(31,167,100,146),(32,168,101,147),(33,157,102,148),(34,158,103,149),(35,159,104,150),(36,160,105,151),(121,175,187,136),(122,176,188,137),(123,177,189,138),(124,178,190,139),(125,179,191,140),(126,180,192,141),(127,169,181,142),(128,170,182,143),(129,171,183,144),(130,172,184,133),(131,173,185,134),(132,174,186,135)], [(1,109,70,44),(2,110,71,45),(3,111,72,46),(4,112,61,47),(5,113,62,48),(6,114,63,37),(7,115,64,38),(8,116,65,39),(9,117,66,40),(10,118,67,41),(11,119,68,42),(12,120,69,43),(13,96,58,77),(14,85,59,78),(15,86,60,79),(16,87,49,80),(17,88,50,81),(18,89,51,82),(19,90,52,83),(20,91,53,84),(21,92,54,73),(22,93,55,74),(23,94,56,75),(24,95,57,76),(25,138,106,177),(26,139,107,178),(27,140,108,179),(28,141,97,180),(29,142,98,169),(30,143,99,170),(31,144,100,171),(32,133,101,172),(33,134,102,173),(34,135,103,174),(35,136,104,175),(36,137,105,176),(121,159,187,150),(122,160,188,151),(123,161,189,152),(124,162,190,153),(125,163,191,154),(126,164,192,155),(127,165,181,156),(128,166,182,145),(129,167,183,146),(130,168,184,147),(131,157,185,148),(132,158,186,149)], [(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)], [(1,176,64,143),(2,136,65,169),(3,174,66,141),(4,134,67,179),(5,172,68,139),(6,144,69,177),(7,170,70,137),(8,142,71,175),(9,180,72,135),(10,140,61,173),(11,178,62,133),(12,138,63,171),(13,25,52,100),(14,105,53,30),(15,35,54,98),(16,103,55,28),(17,33,56,108),(18,101,57,26),(19,31,58,106),(20,99,59,36),(21,29,60,104),(22,97,49,34),(23,27,50,102),(24,107,51,32),(37,167,120,152),(38,145,109,160),(39,165,110,150),(40,155,111,158),(41,163,112,148),(42,153,113,168),(43,161,114,146),(44,151,115,166),(45,159,116,156),(46,149,117,164),(47,157,118,154),(48,147,119,162),(73,181,86,121),(74,126,87,186),(75,191,88,131),(76,124,89,184),(77,189,90,129),(78,122,91,182),(79,187,92,127),(80,132,93,192),(81,185,94,125),(82,130,95,190),(83,183,96,123),(84,128,85,188)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D8 | C4○D12 | C8.C22 | Q8.11D6 | Q8.13D6 |
| kernel | Q8.5Dic6 | C12⋊C8 | C6.Q16 | C12.Q8 | Q8⋊2Dic3 | C12.6Q8 | Q8×C12 | C4×Q8 | C2×C12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C2×C4 | Q8 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of Q8.5Dic6 ►in GL4(𝔽73) generated by
| 72 | 3 | 0 | 0 |
| 48 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 36 | 25 | 0 | 0 |
| 62 | 37 | 0 | 0 |
| 0 | 0 | 43 | 13 |
| 0 | 0 | 60 | 30 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 14 | 7 |
| 0 | 0 | 66 | 7 |
| 52 | 64 | 0 | 0 |
| 57 | 21 | 0 | 0 |
| 0 | 0 | 8 | 47 |
| 0 | 0 | 39 | 65 |
G:=sub<GL(4,GF(73))| [72,48,0,0,3,1,0,0,0,0,72,0,0,0,0,72],[36,62,0,0,25,37,0,0,0,0,43,60,0,0,13,30],[46,0,0,0,0,46,0,0,0,0,14,66,0,0,7,7],[52,57,0,0,64,21,0,0,0,0,8,39,0,0,47,65] >;
Q8.5Dic6 in GAP, Magma, Sage, TeX
Q_8._5{\rm Dic}_6 % in TeX
G:=Group("Q8.5Dic6"); // GroupNames label
G:=SmallGroup(192,581);
// by ID
G=gap.SmallGroup(192,581);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=a^2*c^6,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations