direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C8⋊C4, C42.5C12, C12.37C42, C8⋊9(C2×C12), (C2×C8)⋊8C12, (C2×C24)⋊18C4, C24⋊35(C2×C4), C4.8(C4×C12), (C4×C12).8C4, (C2×C42).9C6, (C22×C8).16C6, C6.33(C2×C42), C22.9(C4×C12), (C2×C6).31C42, C42.57(C2×C6), C2.1(C6×M4(2)), (C22×C12).17C4, C23.42(C2×C12), C4.32(C22×C12), (C22×C24).32C2, (C22×C4).12C12, (C2×C6).29M4(2), C6.45(C2×M4(2)), C12.190(C22×C4), (C2×C12).978C23, (C2×C24).444C22, (C4×C12).298C22, C22.8(C3×M4(2)), C22.16(C22×C12), (C22×C12).607C22, (C2×C4×C12).8C2, C2.5(C2×C4×C12), (C2×C8).98(C2×C6), (C2×C4).58(C2×C12), (C2×C12).287(C2×C4), (C2×C6).228(C22×C4), (C22×C6).146(C2×C4), (C2×C4).146(C22×C6), (C22×C4).141(C2×C6), SmallGroup(192,836)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C8⋊C4
G = < a,b,c | a6=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 162 in 146 conjugacy classes, 130 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C8⋊C4, C2×C42, C22×C8, C4×C12, C2×C24, C22×C12, C22×C12, C2×C8⋊C4, C3×C8⋊C4, C2×C4×C12, C22×C24, C6×C8⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, M4(2), C22×C4, C2×C12, C22×C6, C8⋊C4, C2×C42, C2×M4(2), C4×C12, C3×M4(2), C22×C12, C2×C8⋊C4, C3×C8⋊C4, C2×C4×C12, C6×M4(2), C6×C8⋊C4
►Regular action on 192 points
Generators in S192
(1 191 98 45 50 148)(2 192 99 46 51 149)(3 185 100 47 52 150)(4 186 101 48 53 151)(5 187 102 41 54 152)(6 188 103 42 55 145)(7 189 104 43 56 146)(8 190 97 44 49 147)(9 80 139 117 155 89)(10 73 140 118 156 90)(11 74 141 119 157 91)(12 75 142 120 158 92)(13 76 143 113 159 93)(14 77 144 114 160 94)(15 78 137 115 153 95)(16 79 138 116 154 96)(17 81 25 121 163 129)(18 82 26 122 164 130)(19 83 27 123 165 131)(20 84 28 124 166 132)(21 85 29 125 167 133)(22 86 30 126 168 134)(23 87 31 127 161 135)(24 88 32 128 162 136)(33 70 171 110 179 62)(34 71 172 111 180 63)(35 72 173 112 181 64)(36 65 174 105 182 57)(37 66 175 106 183 58)(38 67 176 107 184 59)(39 68 169 108 177 60)(40 69 170 109 178 61)
(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 93 171 17)(2 90 172 22)(3 95 173 19)(4 92 174 24)(5 89 175 21)(6 94 176 18)(7 91 169 23)(8 96 170 20)(9 106 85 187)(10 111 86 192)(11 108 87 189)(12 105 88 186)(13 110 81 191)(14 107 82 188)(15 112 83 185)(16 109 84 190)(25 98 76 179)(26 103 77 184)(27 100 78 181)(28 97 79 178)(29 102 80 183)(30 99 73 180)(31 104 74 177)(32 101 75 182)(33 163 50 113)(34 168 51 118)(35 165 52 115)(36 162 53 120)(37 167 54 117)(38 164 55 114)(39 161 56 119)(40 166 49 116)(41 139 58 125)(42 144 59 122)(43 141 60 127)(44 138 61 124)(45 143 62 121)(46 140 63 126)(47 137 64 123)(48 142 57 128)(65 136 151 158)(66 133 152 155)(67 130 145 160)(68 135 146 157)(69 132 147 154)(70 129 148 159)(71 134 149 156)(72 131 150 153)
G:=sub<Sym(192)| (1,191,98,45,50,148)(2,192,99,46,51,149)(3,185,100,47,52,150)(4,186,101,48,53,151)(5,187,102,41,54,152)(6,188,103,42,55,145)(7,189,104,43,56,146)(8,190,97,44,49,147)(9,80,139,117,155,89)(10,73,140,118,156,90)(11,74,141,119,157,91)(12,75,142,120,158,92)(13,76,143,113,159,93)(14,77,144,114,160,94)(15,78,137,115,153,95)(16,79,138,116,154,96)(17,81,25,121,163,129)(18,82,26,122,164,130)(19,83,27,123,165,131)(20,84,28,124,166,132)(21,85,29,125,167,133)(22,86,30,126,168,134)(23,87,31,127,161,135)(24,88,32,128,162,136)(33,70,171,110,179,62)(34,71,172,111,180,63)(35,72,173,112,181,64)(36,65,174,105,182,57)(37,66,175,106,183,58)(38,67,176,107,184,59)(39,68,169,108,177,60)(40,69,170,109,178,61), (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), (1,93,171,17)(2,90,172,22)(3,95,173,19)(4,92,174,24)(5,89,175,21)(6,94,176,18)(7,91,169,23)(8,96,170,20)(9,106,85,187)(10,111,86,192)(11,108,87,189)(12,105,88,186)(13,110,81,191)(14,107,82,188)(15,112,83,185)(16,109,84,190)(25,98,76,179)(26,103,77,184)(27,100,78,181)(28,97,79,178)(29,102,80,183)(30,99,73,180)(31,104,74,177)(32,101,75,182)(33,163,50,113)(34,168,51,118)(35,165,52,115)(36,162,53,120)(37,167,54,117)(38,164,55,114)(39,161,56,119)(40,166,49,116)(41,139,58,125)(42,144,59,122)(43,141,60,127)(44,138,61,124)(45,143,62,121)(46,140,63,126)(47,137,64,123)(48,142,57,128)(65,136,151,158)(66,133,152,155)(67,130,145,160)(68,135,146,157)(69,132,147,154)(70,129,148,159)(71,134,149,156)(72,131,150,153)>;
G:=Group( (1,191,98,45,50,148)(2,192,99,46,51,149)(3,185,100,47,52,150)(4,186,101,48,53,151)(5,187,102,41,54,152)(6,188,103,42,55,145)(7,189,104,43,56,146)(8,190,97,44,49,147)(9,80,139,117,155,89)(10,73,140,118,156,90)(11,74,141,119,157,91)(12,75,142,120,158,92)(13,76,143,113,159,93)(14,77,144,114,160,94)(15,78,137,115,153,95)(16,79,138,116,154,96)(17,81,25,121,163,129)(18,82,26,122,164,130)(19,83,27,123,165,131)(20,84,28,124,166,132)(21,85,29,125,167,133)(22,86,30,126,168,134)(23,87,31,127,161,135)(24,88,32,128,162,136)(33,70,171,110,179,62)(34,71,172,111,180,63)(35,72,173,112,181,64)(36,65,174,105,182,57)(37,66,175,106,183,58)(38,67,176,107,184,59)(39,68,169,108,177,60)(40,69,170,109,178,61), (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), (1,93,171,17)(2,90,172,22)(3,95,173,19)(4,92,174,24)(5,89,175,21)(6,94,176,18)(7,91,169,23)(8,96,170,20)(9,106,85,187)(10,111,86,192)(11,108,87,189)(12,105,88,186)(13,110,81,191)(14,107,82,188)(15,112,83,185)(16,109,84,190)(25,98,76,179)(26,103,77,184)(27,100,78,181)(28,97,79,178)(29,102,80,183)(30,99,73,180)(31,104,74,177)(32,101,75,182)(33,163,50,113)(34,168,51,118)(35,165,52,115)(36,162,53,120)(37,167,54,117)(38,164,55,114)(39,161,56,119)(40,166,49,116)(41,139,58,125)(42,144,59,122)(43,141,60,127)(44,138,61,124)(45,143,62,121)(46,140,63,126)(47,137,64,123)(48,142,57,128)(65,136,151,158)(66,133,152,155)(67,130,145,160)(68,135,146,157)(69,132,147,154)(70,129,148,159)(71,134,149,156)(72,131,150,153) );
G=PermutationGroup([[(1,191,98,45,50,148),(2,192,99,46,51,149),(3,185,100,47,52,150),(4,186,101,48,53,151),(5,187,102,41,54,152),(6,188,103,42,55,145),(7,189,104,43,56,146),(8,190,97,44,49,147),(9,80,139,117,155,89),(10,73,140,118,156,90),(11,74,141,119,157,91),(12,75,142,120,158,92),(13,76,143,113,159,93),(14,77,144,114,160,94),(15,78,137,115,153,95),(16,79,138,116,154,96),(17,81,25,121,163,129),(18,82,26,122,164,130),(19,83,27,123,165,131),(20,84,28,124,166,132),(21,85,29,125,167,133),(22,86,30,126,168,134),(23,87,31,127,161,135),(24,88,32,128,162,136),(33,70,171,110,179,62),(34,71,172,111,180,63),(35,72,173,112,181,64),(36,65,174,105,182,57),(37,66,175,106,183,58),(38,67,176,107,184,59),(39,68,169,108,177,60),(40,69,170,109,178,61)], [(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)], [(1,93,171,17),(2,90,172,22),(3,95,173,19),(4,92,174,24),(5,89,175,21),(6,94,176,18),(7,91,169,23),(8,96,170,20),(9,106,85,187),(10,111,86,192),(11,108,87,189),(12,105,88,186),(13,110,81,191),(14,107,82,188),(15,112,83,185),(16,109,84,190),(25,98,76,179),(26,103,77,184),(27,100,78,181),(28,97,79,178),(29,102,80,183),(30,99,73,180),(31,104,74,177),(32,101,75,182),(33,163,50,113),(34,168,51,118),(35,165,52,115),(36,162,53,120),(37,167,54,117),(38,164,55,114),(39,161,56,119),(40,166,49,116),(41,139,58,125),(42,144,59,122),(43,141,60,127),(44,138,61,124),(45,143,62,121),(46,140,63,126),(47,137,64,123),(48,142,57,128),(65,136,151,158),(66,133,152,155),(67,130,145,160),(68,135,146,157),(69,132,147,154),(70,129,148,159),(71,134,149,156),(72,131,150,153)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 8A | ··· | 8P | 12A | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | C6×C8⋊C4 | C3×C8⋊C4 | C2×C4×C12 | C22×C24 | C2×C8⋊C4 | C4×C12 | C2×C24 | C22×C12 | C8⋊C4 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C2×C6 | C22 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 16 | 4 | 8 | 2 | 4 | 8 | 32 | 8 | 8 | 16 |
Matrix representation of C6×C8⋊C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 65 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 42 |
| 0 | 0 | 63 | 3 |
| 27 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 43 | 2 |
| 0 | 0 | 24 | 30 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,72,0,0,0,0,65,0,0,0,0,65],[72,0,0,0,0,1,0,0,0,0,70,63,0,0,42,3],[27,0,0,0,0,72,0,0,0,0,43,24,0,0,2,30] >;
C6×C8⋊C4 in GAP, Magma, Sage, TeX
C_6\times C_8\rtimes C_4
% in TeX
G:=Group("C6xC8:C4"); // GroupNames label
G:=SmallGroup(192,836);
// by ID
G=gap.SmallGroup(192,836);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,1373,344,172]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations