direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8.5Q8, C24.21Q8, (C4×C8).9C6, C8.4(C3×Q8), C4.6(C6×Q8), C2.D8.6C6, C4.Q8.7C6, (C4×C24).20C2, C6.41(C4⋊Q8), C12.95(C2×Q8), (C2×C12).368D4, C42.84(C2×C6), C42.C2.4C6, C6.132(C4○D8), C22.120(C6×D4), (C2×C24).441C22, (C4×C12).368C22, (C2×C12).955C23, C2.7(C3×C4⋊Q8), C4⋊C4.24(C2×C6), (C2×C8).82(C2×C6), C2.19(C3×C4○D8), (C2×C4).58(C3×D4), (C2×C6).676(C2×D4), (C3×C4.Q8).14C2, (C3×C2.D8).15C2, (C3×C4⋊C4).244C22, (C2×C4).130(C22×C6), (C3×C42.C2).11C2, SmallGroup(192,932)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.5Q8
G = < a,b,c,d | a3=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
Subgroups: 130 in 86 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C24, C2×C12, C2×C12, C2×C12, C4×C8, C4.Q8, C2.D8, C42.C2, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C8.5Q8, C4×C24, C3×C4.Q8, C3×C2.D8, C3×C42.C2, C3×C8.5Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C4○D8, C6×D4, C6×Q8, C8.5Q8, C3×C4⋊Q8, C3×C4○D8, C3×C8.5Q8
►Regular action on 192 points
Generators in S192
(1 16 33)(2 9 34)(3 10 35)(4 11 36)(5 12 37)(6 13 38)(7 14 39)(8 15 40)(17 49 41)(18 50 42)(19 51 43)(20 52 44)(21 53 45)(22 54 46)(23 55 47)(24 56 48)(25 189 163)(26 190 164)(27 191 165)(28 192 166)(29 185 167)(30 186 168)(31 187 161)(32 188 162)(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)(105 128 131)(106 121 132)(107 122 133)(108 123 134)(109 124 135)(110 125 136)(111 126 129)(112 127 130)(113 145 139)(114 146 140)(115 147 141)(116 148 142)(117 149 143)(118 150 144)(119 151 137)(120 152 138)(153 176 179)(154 169 180)(155 170 181)(156 171 182)(157 172 183)(158 173 184)(159 174 177)(160 175 178)
(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 81)(2 68 20 82)(3 69 21 83)(4 70 22 84)(5 71 23 85)(6 72 24 86)(7 65 17 87)(8 66 18 88)(9 98 52 57)(10 99 53 58)(11 100 54 59)(12 101 55 60)(13 102 56 61)(14 103 49 62)(15 104 50 63)(16 97 51 64)(25 126 170 151)(26 127 171 152)(27 128 172 145)(28 121 173 146)(29 122 174 147)(30 123 175 148)(31 124 176 149)(32 125 169 150)(33 91 43 77)(34 92 44 78)(35 93 45 79)(36 94 46 80)(37 95 47 73)(38 96 48 74)(39 89 41 75)(40 90 42 76)(105 157 113 165)(106 158 114 166)(107 159 115 167)(108 160 116 168)(109 153 117 161)(110 154 118 162)(111 155 119 163)(112 156 120 164)(129 181 137 189)(130 182 138 190)(131 183 139 191)(132 184 140 192)(133 177 141 185)(134 178 142 186)(135 179 143 187)(136 180 144 188)
(1 115 23 111)(2 118 24 106)(3 113 17 109)(4 116 18 112)(5 119 19 107)(6 114 20 110)(7 117 21 105)(8 120 22 108)(9 150 56 121)(10 145 49 124)(11 148 50 127)(12 151 51 122)(13 146 52 125)(14 149 53 128)(15 152 54 123)(16 147 55 126)(25 101 174 64)(26 104 175 59)(27 99 176 62)(28 102 169 57)(29 97 170 60)(30 100 171 63)(31 103 172 58)(32 98 173 61)(33 141 47 129)(34 144 48 132)(35 139 41 135)(36 142 42 130)(37 137 43 133)(38 140 44 136)(39 143 45 131)(40 138 46 134)(65 157 83 161)(66 160 84 164)(67 155 85 167)(68 158 86 162)(69 153 87 165)(70 156 88 168)(71 159 81 163)(72 154 82 166)(73 185 91 181)(74 188 92 184)(75 191 93 179)(76 186 94 182)(77 189 95 177)(78 192 96 180)(79 187 89 183)(80 190 90 178)
G:=sub<Sym(192)| (1,16,33)(2,9,34)(3,10,35)(4,11,36)(5,12,37)(6,13,38)(7,14,39)(8,15,40)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,189,163)(26,190,164)(27,191,165)(28,192,166)(29,185,167)(30,186,168)(31,187,161)(32,188,162)(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)(105,128,131)(106,121,132)(107,122,133)(108,123,134)(109,124,135)(110,125,136)(111,126,129)(112,127,130)(113,145,139)(114,146,140)(115,147,141)(116,148,142)(117,149,143)(118,150,144)(119,151,137)(120,152,138)(153,176,179)(154,169,180)(155,170,181)(156,171,182)(157,172,183)(158,173,184)(159,174,177)(160,175,178), (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), (1,67,19,81)(2,68,20,82)(3,69,21,83)(4,70,22,84)(5,71,23,85)(6,72,24,86)(7,65,17,87)(8,66,18,88)(9,98,52,57)(10,99,53,58)(11,100,54,59)(12,101,55,60)(13,102,56,61)(14,103,49,62)(15,104,50,63)(16,97,51,64)(25,126,170,151)(26,127,171,152)(27,128,172,145)(28,121,173,146)(29,122,174,147)(30,123,175,148)(31,124,176,149)(32,125,169,150)(33,91,43,77)(34,92,44,78)(35,93,45,79)(36,94,46,80)(37,95,47,73)(38,96,48,74)(39,89,41,75)(40,90,42,76)(105,157,113,165)(106,158,114,166)(107,159,115,167)(108,160,116,168)(109,153,117,161)(110,154,118,162)(111,155,119,163)(112,156,120,164)(129,181,137,189)(130,182,138,190)(131,183,139,191)(132,184,140,192)(133,177,141,185)(134,178,142,186)(135,179,143,187)(136,180,144,188), (1,115,23,111)(2,118,24,106)(3,113,17,109)(4,116,18,112)(5,119,19,107)(6,114,20,110)(7,117,21,105)(8,120,22,108)(9,150,56,121)(10,145,49,124)(11,148,50,127)(12,151,51,122)(13,146,52,125)(14,149,53,128)(15,152,54,123)(16,147,55,126)(25,101,174,64)(26,104,175,59)(27,99,176,62)(28,102,169,57)(29,97,170,60)(30,100,171,63)(31,103,172,58)(32,98,173,61)(33,141,47,129)(34,144,48,132)(35,139,41,135)(36,142,42,130)(37,137,43,133)(38,140,44,136)(39,143,45,131)(40,138,46,134)(65,157,83,161)(66,160,84,164)(67,155,85,167)(68,158,86,162)(69,153,87,165)(70,156,88,168)(71,159,81,163)(72,154,82,166)(73,185,91,181)(74,188,92,184)(75,191,93,179)(76,186,94,182)(77,189,95,177)(78,192,96,180)(79,187,89,183)(80,190,90,178)>;
G:=Group( (1,16,33)(2,9,34)(3,10,35)(4,11,36)(5,12,37)(6,13,38)(7,14,39)(8,15,40)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,189,163)(26,190,164)(27,191,165)(28,192,166)(29,185,167)(30,186,168)(31,187,161)(32,188,162)(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)(105,128,131)(106,121,132)(107,122,133)(108,123,134)(109,124,135)(110,125,136)(111,126,129)(112,127,130)(113,145,139)(114,146,140)(115,147,141)(116,148,142)(117,149,143)(118,150,144)(119,151,137)(120,152,138)(153,176,179)(154,169,180)(155,170,181)(156,171,182)(157,172,183)(158,173,184)(159,174,177)(160,175,178), (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), (1,67,19,81)(2,68,20,82)(3,69,21,83)(4,70,22,84)(5,71,23,85)(6,72,24,86)(7,65,17,87)(8,66,18,88)(9,98,52,57)(10,99,53,58)(11,100,54,59)(12,101,55,60)(13,102,56,61)(14,103,49,62)(15,104,50,63)(16,97,51,64)(25,126,170,151)(26,127,171,152)(27,128,172,145)(28,121,173,146)(29,122,174,147)(30,123,175,148)(31,124,176,149)(32,125,169,150)(33,91,43,77)(34,92,44,78)(35,93,45,79)(36,94,46,80)(37,95,47,73)(38,96,48,74)(39,89,41,75)(40,90,42,76)(105,157,113,165)(106,158,114,166)(107,159,115,167)(108,160,116,168)(109,153,117,161)(110,154,118,162)(111,155,119,163)(112,156,120,164)(129,181,137,189)(130,182,138,190)(131,183,139,191)(132,184,140,192)(133,177,141,185)(134,178,142,186)(135,179,143,187)(136,180,144,188), (1,115,23,111)(2,118,24,106)(3,113,17,109)(4,116,18,112)(5,119,19,107)(6,114,20,110)(7,117,21,105)(8,120,22,108)(9,150,56,121)(10,145,49,124)(11,148,50,127)(12,151,51,122)(13,146,52,125)(14,149,53,128)(15,152,54,123)(16,147,55,126)(25,101,174,64)(26,104,175,59)(27,99,176,62)(28,102,169,57)(29,97,170,60)(30,100,171,63)(31,103,172,58)(32,98,173,61)(33,141,47,129)(34,144,48,132)(35,139,41,135)(36,142,42,130)(37,137,43,133)(38,140,44,136)(39,143,45,131)(40,138,46,134)(65,157,83,161)(66,160,84,164)(67,155,85,167)(68,158,86,162)(69,153,87,165)(70,156,88,168)(71,159,81,163)(72,154,82,166)(73,185,91,181)(74,188,92,184)(75,191,93,179)(76,186,94,182)(77,189,95,177)(78,192,96,180)(79,187,89,183)(80,190,90,178) );
G=PermutationGroup([[(1,16,33),(2,9,34),(3,10,35),(4,11,36),(5,12,37),(6,13,38),(7,14,39),(8,15,40),(17,49,41),(18,50,42),(19,51,43),(20,52,44),(21,53,45),(22,54,46),(23,55,47),(24,56,48),(25,189,163),(26,190,164),(27,191,165),(28,192,166),(29,185,167),(30,186,168),(31,187,161),(32,188,162),(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),(105,128,131),(106,121,132),(107,122,133),(108,123,134),(109,124,135),(110,125,136),(111,126,129),(112,127,130),(113,145,139),(114,146,140),(115,147,141),(116,148,142),(117,149,143),(118,150,144),(119,151,137),(120,152,138),(153,176,179),(154,169,180),(155,170,181),(156,171,182),(157,172,183),(158,173,184),(159,174,177),(160,175,178)], [(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)], [(1,67,19,81),(2,68,20,82),(3,69,21,83),(4,70,22,84),(5,71,23,85),(6,72,24,86),(7,65,17,87),(8,66,18,88),(9,98,52,57),(10,99,53,58),(11,100,54,59),(12,101,55,60),(13,102,56,61),(14,103,49,62),(15,104,50,63),(16,97,51,64),(25,126,170,151),(26,127,171,152),(27,128,172,145),(28,121,173,146),(29,122,174,147),(30,123,175,148),(31,124,176,149),(32,125,169,150),(33,91,43,77),(34,92,44,78),(35,93,45,79),(36,94,46,80),(37,95,47,73),(38,96,48,74),(39,89,41,75),(40,90,42,76),(105,157,113,165),(106,158,114,166),(107,159,115,167),(108,160,116,168),(109,153,117,161),(110,154,118,162),(111,155,119,163),(112,156,120,164),(129,181,137,189),(130,182,138,190),(131,183,139,191),(132,184,140,192),(133,177,141,185),(134,178,142,186),(135,179,143,187),(136,180,144,188)], [(1,115,23,111),(2,118,24,106),(3,113,17,109),(4,116,18,112),(5,119,19,107),(6,114,20,110),(7,117,21,105),(8,120,22,108),(9,150,56,121),(10,145,49,124),(11,148,50,127),(12,151,51,122),(13,146,52,125),(14,149,53,128),(15,152,54,123),(16,147,55,126),(25,101,174,64),(26,104,175,59),(27,99,176,62),(28,102,169,57),(29,97,170,60),(30,100,171,63),(31,103,172,58),(32,98,173,61),(33,141,47,129),(34,144,48,132),(35,139,41,135),(36,142,42,130),(37,137,43,133),(38,140,44,136),(39,143,45,131),(40,138,46,134),(65,157,83,161),(66,160,84,164),(67,155,85,167),(68,158,86,162),(69,153,87,165),(70,156,88,168),(71,159,81,163),(72,154,82,166),(73,185,91,181),(74,188,92,184),(75,191,93,179),(76,186,94,182),(77,189,95,177),(78,192,96,180),(79,187,89,183),(80,190,90,178)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | Q8 | D4 | C3×Q8 | C3×D4 | C4○D8 | C3×C4○D8 |
| kernel | C3×C8.5Q8 | C4×C24 | C3×C4.Q8 | C3×C2.D8 | C3×C42.C2 | C8.5Q8 | C4×C8 | C4.Q8 | C2.D8 | C42.C2 | C24 | C2×C12 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 16 |
Matrix representation of C3×C8.5Q8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 71 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 15 | 22 |
| 72 | 71 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 31 | 72 |
| 49 | 34 | 0 | 0 |
| 41 | 24 | 0 | 0 |
| 0 | 0 | 31 | 71 |
| 0 | 0 | 43 | 42 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[72,1,0,0,71,1,0,0,0,0,63,15,0,0,0,22],[72,1,0,0,71,1,0,0,0,0,1,31,0,0,0,72],[49,41,0,0,34,24,0,0,0,0,31,43,0,0,71,42] >;
C3×C8.5Q8 in GAP, Magma, Sage, TeX
C_3\times C_8._5Q_8
% in TeX
G:=Group("C3xC8.5Q8"); // GroupNames label
G:=SmallGroup(192,932);
// by ID
G=gap.SmallGroup(192,932);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,848,1094,268,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations