direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊2Q8, C24⋊10Q8, C12.43D8, C12.21Q16, C8⋊2(C3×Q8), (C4×C8).8C6, C4.5(C3×D8), C4.7(C6×Q8), C6.83(C2×D8), C2.11(C6×D8), C4⋊Q8.11C6, C4.4(C3×Q16), C2.D8.7C6, (C4×C24).19C2, C6.42(C4⋊Q8), C2.11(C6×Q16), C12.96(C2×Q8), C6.58(C2×Q16), (C2×C12).425D4, C42.85(C2×C6), C22.121(C6×D4), (C4×C12).369C22, (C2×C24).403C22, (C2×C12).956C23, C2.8(C3×C4⋊Q8), C4⋊C4.25(C2×C6), (C2×C8).83(C2×C6), (C2×C4).81(C3×D4), (C3×C4⋊Q8).26C2, (C2×C6).677(C2×D4), (C3×C2.D8).16C2, (C3×C4⋊C4).245C22, (C2×C4).131(C22×C6), SmallGroup(192,933)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊2Q8
G = < a,b,c,d | a3=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 162 in 98 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C4×C8, C2.D8, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C8⋊2Q8, C4×C24, C3×C2.D8, C3×C4⋊Q8, C3×C8⋊2Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, D8, Q16, C2×D4, C2×Q8, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C2×D8, C2×Q16, C3×D8, C3×Q16, C6×D4, C6×Q8, C8⋊2Q8, C3×C4⋊Q8, C6×D8, C6×Q16, C3×C8⋊2Q8
►Regular action on 192 points
Generators in S192
(1 35 11)(2 36 12)(3 37 13)(4 38 14)(5 39 15)(6 40 16)(7 33 9)(8 34 10)(17 49 41)(18 50 42)(19 51 43)(20 52 44)(21 53 45)(22 54 46)(23 55 47)(24 56 48)(25 190 164)(26 191 165)(27 192 166)(28 185 167)(29 186 168)(30 187 161)(31 188 162)(32 189 163)(57 78 82)(58 79 83)(59 80 84)(60 73 85)(61 74 86)(62 75 87)(63 76 88)(64 77 81)(65 103 89)(66 104 90)(67 97 91)(68 98 92)(69 99 93)(70 100 94)(71 101 95)(72 102 96)(105 124 131)(106 125 132)(107 126 133)(108 127 134)(109 128 135)(110 121 136)(111 122 129)(112 123 130)(113 145 139)(114 146 140)(115 147 141)(116 148 142)(117 149 143)(118 150 144)(119 151 137)(120 152 138)(153 172 179)(154 173 180)(155 174 181)(156 175 182)(157 176 183)(158 169 184)(159 170 177)(160 171 178)
(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 67 19 83)(2 68 20 84)(3 69 21 85)(4 70 22 86)(5 71 23 87)(6 72 24 88)(7 65 17 81)(8 66 18 82)(9 89 41 77)(10 90 42 78)(11 91 43 79)(12 92 44 80)(13 93 45 73)(14 94 46 74)(15 95 47 75)(16 96 48 76)(25 121 169 148)(26 122 170 149)(27 123 171 150)(28 124 172 151)(29 125 173 152)(30 126 174 145)(31 127 175 146)(32 128 176 147)(33 103 49 64)(34 104 50 57)(35 97 51 58)(36 98 52 59)(37 99 53 60)(38 100 54 61)(39 101 55 62)(40 102 56 63)(105 153 119 167)(106 154 120 168)(107 155 113 161)(108 156 114 162)(109 157 115 163)(110 158 116 164)(111 159 117 165)(112 160 118 166)(129 177 143 191)(130 178 144 192)(131 179 137 185)(132 180 138 186)(133 181 139 187)(134 182 140 188)(135 183 141 189)(136 184 142 190)
(1 115 19 109)(2 114 20 108)(3 113 21 107)(4 120 22 106)(5 119 23 105)(6 118 24 112)(7 117 17 111)(8 116 18 110)(9 143 41 129)(10 142 42 136)(11 141 43 135)(12 140 44 134)(13 139 45 133)(14 138 46 132)(15 137 47 131)(16 144 48 130)(25 104 169 57)(26 103 170 64)(27 102 171 63)(28 101 172 62)(29 100 173 61)(30 99 174 60)(31 98 175 59)(32 97 176 58)(33 149 49 122)(34 148 50 121)(35 147 51 128)(36 146 52 127)(37 145 53 126)(38 152 54 125)(39 151 55 124)(40 150 56 123)(65 159 81 165)(66 158 82 164)(67 157 83 163)(68 156 84 162)(69 155 85 161)(70 154 86 168)(71 153 87 167)(72 160 88 166)(73 187 93 181)(74 186 94 180)(75 185 95 179)(76 192 96 178)(77 191 89 177)(78 190 90 184)(79 189 91 183)(80 188 92 182)
G:=sub<Sym(192)| (1,35,11)(2,36,12)(3,37,13)(4,38,14)(5,39,15)(6,40,16)(7,33,9)(8,34,10)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,190,164)(26,191,165)(27,192,166)(28,185,167)(29,186,168)(30,187,161)(31,188,162)(32,189,163)(57,78,82)(58,79,83)(59,80,84)(60,73,85)(61,74,86)(62,75,87)(63,76,88)(64,77,81)(65,103,89)(66,104,90)(67,97,91)(68,98,92)(69,99,93)(70,100,94)(71,101,95)(72,102,96)(105,124,131)(106,125,132)(107,126,133)(108,127,134)(109,128,135)(110,121,136)(111,122,129)(112,123,130)(113,145,139)(114,146,140)(115,147,141)(116,148,142)(117,149,143)(118,150,144)(119,151,137)(120,152,138)(153,172,179)(154,173,180)(155,174,181)(156,175,182)(157,176,183)(158,169,184)(159,170,177)(160,171,178), (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), (1,67,19,83)(2,68,20,84)(3,69,21,85)(4,70,22,86)(5,71,23,87)(6,72,24,88)(7,65,17,81)(8,66,18,82)(9,89,41,77)(10,90,42,78)(11,91,43,79)(12,92,44,80)(13,93,45,73)(14,94,46,74)(15,95,47,75)(16,96,48,76)(25,121,169,148)(26,122,170,149)(27,123,171,150)(28,124,172,151)(29,125,173,152)(30,126,174,145)(31,127,175,146)(32,128,176,147)(33,103,49,64)(34,104,50,57)(35,97,51,58)(36,98,52,59)(37,99,53,60)(38,100,54,61)(39,101,55,62)(40,102,56,63)(105,153,119,167)(106,154,120,168)(107,155,113,161)(108,156,114,162)(109,157,115,163)(110,158,116,164)(111,159,117,165)(112,160,118,166)(129,177,143,191)(130,178,144,192)(131,179,137,185)(132,180,138,186)(133,181,139,187)(134,182,140,188)(135,183,141,189)(136,184,142,190), (1,115,19,109)(2,114,20,108)(3,113,21,107)(4,120,22,106)(5,119,23,105)(6,118,24,112)(7,117,17,111)(8,116,18,110)(9,143,41,129)(10,142,42,136)(11,141,43,135)(12,140,44,134)(13,139,45,133)(14,138,46,132)(15,137,47,131)(16,144,48,130)(25,104,169,57)(26,103,170,64)(27,102,171,63)(28,101,172,62)(29,100,173,61)(30,99,174,60)(31,98,175,59)(32,97,176,58)(33,149,49,122)(34,148,50,121)(35,147,51,128)(36,146,52,127)(37,145,53,126)(38,152,54,125)(39,151,55,124)(40,150,56,123)(65,159,81,165)(66,158,82,164)(67,157,83,163)(68,156,84,162)(69,155,85,161)(70,154,86,168)(71,153,87,167)(72,160,88,166)(73,187,93,181)(74,186,94,180)(75,185,95,179)(76,192,96,178)(77,191,89,177)(78,190,90,184)(79,189,91,183)(80,188,92,182)>;
G:=Group( (1,35,11)(2,36,12)(3,37,13)(4,38,14)(5,39,15)(6,40,16)(7,33,9)(8,34,10)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,190,164)(26,191,165)(27,192,166)(28,185,167)(29,186,168)(30,187,161)(31,188,162)(32,189,163)(57,78,82)(58,79,83)(59,80,84)(60,73,85)(61,74,86)(62,75,87)(63,76,88)(64,77,81)(65,103,89)(66,104,90)(67,97,91)(68,98,92)(69,99,93)(70,100,94)(71,101,95)(72,102,96)(105,124,131)(106,125,132)(107,126,133)(108,127,134)(109,128,135)(110,121,136)(111,122,129)(112,123,130)(113,145,139)(114,146,140)(115,147,141)(116,148,142)(117,149,143)(118,150,144)(119,151,137)(120,152,138)(153,172,179)(154,173,180)(155,174,181)(156,175,182)(157,176,183)(158,169,184)(159,170,177)(160,171,178), (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), (1,67,19,83)(2,68,20,84)(3,69,21,85)(4,70,22,86)(5,71,23,87)(6,72,24,88)(7,65,17,81)(8,66,18,82)(9,89,41,77)(10,90,42,78)(11,91,43,79)(12,92,44,80)(13,93,45,73)(14,94,46,74)(15,95,47,75)(16,96,48,76)(25,121,169,148)(26,122,170,149)(27,123,171,150)(28,124,172,151)(29,125,173,152)(30,126,174,145)(31,127,175,146)(32,128,176,147)(33,103,49,64)(34,104,50,57)(35,97,51,58)(36,98,52,59)(37,99,53,60)(38,100,54,61)(39,101,55,62)(40,102,56,63)(105,153,119,167)(106,154,120,168)(107,155,113,161)(108,156,114,162)(109,157,115,163)(110,158,116,164)(111,159,117,165)(112,160,118,166)(129,177,143,191)(130,178,144,192)(131,179,137,185)(132,180,138,186)(133,181,139,187)(134,182,140,188)(135,183,141,189)(136,184,142,190), (1,115,19,109)(2,114,20,108)(3,113,21,107)(4,120,22,106)(5,119,23,105)(6,118,24,112)(7,117,17,111)(8,116,18,110)(9,143,41,129)(10,142,42,136)(11,141,43,135)(12,140,44,134)(13,139,45,133)(14,138,46,132)(15,137,47,131)(16,144,48,130)(25,104,169,57)(26,103,170,64)(27,102,171,63)(28,101,172,62)(29,100,173,61)(30,99,174,60)(31,98,175,59)(32,97,176,58)(33,149,49,122)(34,148,50,121)(35,147,51,128)(36,146,52,127)(37,145,53,126)(38,152,54,125)(39,151,55,124)(40,150,56,123)(65,159,81,165)(66,158,82,164)(67,157,83,163)(68,156,84,162)(69,155,85,161)(70,154,86,168)(71,153,87,167)(72,160,88,166)(73,187,93,181)(74,186,94,180)(75,185,95,179)(76,192,96,178)(77,191,89,177)(78,190,90,184)(79,189,91,183)(80,188,92,182) );
G=PermutationGroup([[(1,35,11),(2,36,12),(3,37,13),(4,38,14),(5,39,15),(6,40,16),(7,33,9),(8,34,10),(17,49,41),(18,50,42),(19,51,43),(20,52,44),(21,53,45),(22,54,46),(23,55,47),(24,56,48),(25,190,164),(26,191,165),(27,192,166),(28,185,167),(29,186,168),(30,187,161),(31,188,162),(32,189,163),(57,78,82),(58,79,83),(59,80,84),(60,73,85),(61,74,86),(62,75,87),(63,76,88),(64,77,81),(65,103,89),(66,104,90),(67,97,91),(68,98,92),(69,99,93),(70,100,94),(71,101,95),(72,102,96),(105,124,131),(106,125,132),(107,126,133),(108,127,134),(109,128,135),(110,121,136),(111,122,129),(112,123,130),(113,145,139),(114,146,140),(115,147,141),(116,148,142),(117,149,143),(118,150,144),(119,151,137),(120,152,138),(153,172,179),(154,173,180),(155,174,181),(156,175,182),(157,176,183),(158,169,184),(159,170,177),(160,171,178)], [(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)], [(1,67,19,83),(2,68,20,84),(3,69,21,85),(4,70,22,86),(5,71,23,87),(6,72,24,88),(7,65,17,81),(8,66,18,82),(9,89,41,77),(10,90,42,78),(11,91,43,79),(12,92,44,80),(13,93,45,73),(14,94,46,74),(15,95,47,75),(16,96,48,76),(25,121,169,148),(26,122,170,149),(27,123,171,150),(28,124,172,151),(29,125,173,152),(30,126,174,145),(31,127,175,146),(32,128,176,147),(33,103,49,64),(34,104,50,57),(35,97,51,58),(36,98,52,59),(37,99,53,60),(38,100,54,61),(39,101,55,62),(40,102,56,63),(105,153,119,167),(106,154,120,168),(107,155,113,161),(108,156,114,162),(109,157,115,163),(110,158,116,164),(111,159,117,165),(112,160,118,166),(129,177,143,191),(130,178,144,192),(131,179,137,185),(132,180,138,186),(133,181,139,187),(134,182,140,188),(135,183,141,189),(136,184,142,190)], [(1,115,19,109),(2,114,20,108),(3,113,21,107),(4,120,22,106),(5,119,23,105),(6,118,24,112),(7,117,17,111),(8,116,18,110),(9,143,41,129),(10,142,42,136),(11,141,43,135),(12,140,44,134),(13,139,45,133),(14,138,46,132),(15,137,47,131),(16,144,48,130),(25,104,169,57),(26,103,170,64),(27,102,171,63),(28,101,172,62),(29,100,173,61),(30,99,174,60),(31,98,175,59),(32,97,176,58),(33,149,49,122),(34,148,50,121),(35,147,51,128),(36,146,52,127),(37,145,53,126),(38,152,54,125),(39,151,55,124),(40,150,56,123),(65,159,81,165),(66,158,82,164),(67,157,83,163),(68,156,84,162),(69,155,85,161),(70,154,86,168),(71,153,87,167),(72,160,88,166),(73,187,93,181),(74,186,94,180),(75,185,95,179),(76,192,96,178),(77,191,89,177),(78,190,90,184),(79,189,91,183),(80,188,92,182)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12L | 12M | ··· | 12T | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8 | D4 | D8 | Q16 | C3×Q8 | C3×D4 | C3×D8 | C3×Q16 |
| kernel | C3×C8⋊2Q8 | C4×C24 | C3×C2.D8 | C3×C4⋊Q8 | C8⋊2Q8 | C4×C8 | C2.D8 | C4⋊Q8 | C24 | C2×C12 | C12 | C12 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 8 |
Matrix representation of C3×C8⋊2Q8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 61 | 0 | 0 |
| 61 | 72 | 0 | 0 |
| 0 | 0 | 17 | 2 |
| 0 | 0 | 2 | 56 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,57,57,0,0,16,57],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[1,61,0,0,61,72,0,0,0,0,17,2,0,0,2,56] >;
C3×C8⋊2Q8 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_2Q_8
% in TeX
G:=Group("C3xC8:2Q8"); // GroupNames label
G:=SmallGroup(192,933);
// by ID
G=gap.SmallGroup(192,933);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,176,1094,772,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations