metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.6D16, C6.3Q32, C24.2Q8, C12.3Q16, C8.4Dic6, C3⋊C16⋊5C4, C8.24(C4×S3), C24.7(C2×C4), (C2×C6).31D8, C3⋊1(C16⋊3C4), C12.2(C4⋊C4), C2.D8.1S3, (C2×C12).89D4, (C2×C8).219D6, C6.2(C2.D8), C2.1(C3⋊D16), C24⋊1C4.11C2, C2.1(C3⋊Q32), C4.1(C3⋊Q16), C4.2(Dic3⋊C4), (C2×C24).71C22, C2.3(C6.Q16), C22.12(D4⋊S3), (C2×C3⋊C16).2C2, (C3×C2.D8).1C2, (C2×C4).113(C3⋊D4), SmallGroup(192,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.6D16
G = < a,b,c | a6=b16=1, c2=a3, bab-1=cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C6.6D16
►Regular action on 192 points
Generators in S192
(1 107 135 96 150 64)(2 49 151 81 136 108)(3 109 137 82 152 50)(4 51 153 83 138 110)(5 111 139 84 154 52)(6 53 155 85 140 112)(7 97 141 86 156 54)(8 55 157 87 142 98)(9 99 143 88 158 56)(10 57 159 89 144 100)(11 101 129 90 160 58)(12 59 145 91 130 102)(13 103 131 92 146 60)(14 61 147 93 132 104)(15 105 133 94 148 62)(16 63 149 95 134 106)(17 71 116 42 173 180)(18 181 174 43 117 72)(19 73 118 44 175 182)(20 183 176 45 119 74)(21 75 120 46 161 184)(22 185 162 47 121 76)(23 77 122 48 163 186)(24 187 164 33 123 78)(25 79 124 34 165 188)(26 189 166 35 125 80)(27 65 126 36 167 190)(28 191 168 37 127 66)(29 67 128 38 169 192)(30 177 170 39 113 68)(31 69 114 40 171 178)(32 179 172 41 115 70)
(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)
(1 117 96 181)(2 116 81 180)(3 115 82 179)(4 114 83 178)(5 113 84 177)(6 128 85 192)(7 127 86 191)(8 126 87 190)(9 125 88 189)(10 124 89 188)(11 123 90 187)(12 122 91 186)(13 121 92 185)(14 120 93 184)(15 119 94 183)(16 118 95 182)(17 108 42 151)(18 107 43 150)(19 106 44 149)(20 105 45 148)(21 104 46 147)(22 103 47 146)(23 102 48 145)(24 101 33 160)(25 100 34 159)(26 99 35 158)(27 98 36 157)(28 97 37 156)(29 112 38 155)(30 111 39 154)(31 110 40 153)(32 109 41 152)(49 71 136 173)(50 70 137 172)(51 69 138 171)(52 68 139 170)(53 67 140 169)(54 66 141 168)(55 65 142 167)(56 80 143 166)(57 79 144 165)(58 78 129 164)(59 77 130 163)(60 76 131 162)(61 75 132 161)(62 74 133 176)(63 73 134 175)(64 72 135 174)
G:=sub<Sym(192)| (1,107,135,96,150,64)(2,49,151,81,136,108)(3,109,137,82,152,50)(4,51,153,83,138,110)(5,111,139,84,154,52)(6,53,155,85,140,112)(7,97,141,86,156,54)(8,55,157,87,142,98)(9,99,143,88,158,56)(10,57,159,89,144,100)(11,101,129,90,160,58)(12,59,145,91,130,102)(13,103,131,92,146,60)(14,61,147,93,132,104)(15,105,133,94,148,62)(16,63,149,95,134,106)(17,71,116,42,173,180)(18,181,174,43,117,72)(19,73,118,44,175,182)(20,183,176,45,119,74)(21,75,120,46,161,184)(22,185,162,47,121,76)(23,77,122,48,163,186)(24,187,164,33,123,78)(25,79,124,34,165,188)(26,189,166,35,125,80)(27,65,126,36,167,190)(28,191,168,37,127,66)(29,67,128,38,169,192)(30,177,170,39,113,68)(31,69,114,40,171,178)(32,179,172,41,115,70), (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), (1,117,96,181)(2,116,81,180)(3,115,82,179)(4,114,83,178)(5,113,84,177)(6,128,85,192)(7,127,86,191)(8,126,87,190)(9,125,88,189)(10,124,89,188)(11,123,90,187)(12,122,91,186)(13,121,92,185)(14,120,93,184)(15,119,94,183)(16,118,95,182)(17,108,42,151)(18,107,43,150)(19,106,44,149)(20,105,45,148)(21,104,46,147)(22,103,47,146)(23,102,48,145)(24,101,33,160)(25,100,34,159)(26,99,35,158)(27,98,36,157)(28,97,37,156)(29,112,38,155)(30,111,39,154)(31,110,40,153)(32,109,41,152)(49,71,136,173)(50,70,137,172)(51,69,138,171)(52,68,139,170)(53,67,140,169)(54,66,141,168)(55,65,142,167)(56,80,143,166)(57,79,144,165)(58,78,129,164)(59,77,130,163)(60,76,131,162)(61,75,132,161)(62,74,133,176)(63,73,134,175)(64,72,135,174)>;
G:=Group( (1,107,135,96,150,64)(2,49,151,81,136,108)(3,109,137,82,152,50)(4,51,153,83,138,110)(5,111,139,84,154,52)(6,53,155,85,140,112)(7,97,141,86,156,54)(8,55,157,87,142,98)(9,99,143,88,158,56)(10,57,159,89,144,100)(11,101,129,90,160,58)(12,59,145,91,130,102)(13,103,131,92,146,60)(14,61,147,93,132,104)(15,105,133,94,148,62)(16,63,149,95,134,106)(17,71,116,42,173,180)(18,181,174,43,117,72)(19,73,118,44,175,182)(20,183,176,45,119,74)(21,75,120,46,161,184)(22,185,162,47,121,76)(23,77,122,48,163,186)(24,187,164,33,123,78)(25,79,124,34,165,188)(26,189,166,35,125,80)(27,65,126,36,167,190)(28,191,168,37,127,66)(29,67,128,38,169,192)(30,177,170,39,113,68)(31,69,114,40,171,178)(32,179,172,41,115,70), (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), (1,117,96,181)(2,116,81,180)(3,115,82,179)(4,114,83,178)(5,113,84,177)(6,128,85,192)(7,127,86,191)(8,126,87,190)(9,125,88,189)(10,124,89,188)(11,123,90,187)(12,122,91,186)(13,121,92,185)(14,120,93,184)(15,119,94,183)(16,118,95,182)(17,108,42,151)(18,107,43,150)(19,106,44,149)(20,105,45,148)(21,104,46,147)(22,103,47,146)(23,102,48,145)(24,101,33,160)(25,100,34,159)(26,99,35,158)(27,98,36,157)(28,97,37,156)(29,112,38,155)(30,111,39,154)(31,110,40,153)(32,109,41,152)(49,71,136,173)(50,70,137,172)(51,69,138,171)(52,68,139,170)(53,67,140,169)(54,66,141,168)(55,65,142,167)(56,80,143,166)(57,79,144,165)(58,78,129,164)(59,77,130,163)(60,76,131,162)(61,75,132,161)(62,74,133,176)(63,73,134,175)(64,72,135,174) );
G=PermutationGroup([[(1,107,135,96,150,64),(2,49,151,81,136,108),(3,109,137,82,152,50),(4,51,153,83,138,110),(5,111,139,84,154,52),(6,53,155,85,140,112),(7,97,141,86,156,54),(8,55,157,87,142,98),(9,99,143,88,158,56),(10,57,159,89,144,100),(11,101,129,90,160,58),(12,59,145,91,130,102),(13,103,131,92,146,60),(14,61,147,93,132,104),(15,105,133,94,148,62),(16,63,149,95,134,106),(17,71,116,42,173,180),(18,181,174,43,117,72),(19,73,118,44,175,182),(20,183,176,45,119,74),(21,75,120,46,161,184),(22,185,162,47,121,76),(23,77,122,48,163,186),(24,187,164,33,123,78),(25,79,124,34,165,188),(26,189,166,35,125,80),(27,65,126,36,167,190),(28,191,168,37,127,66),(29,67,128,38,169,192),(30,177,170,39,113,68),(31,69,114,40,171,178),(32,179,172,41,115,70)], [(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)], [(1,117,96,181),(2,116,81,180),(3,115,82,179),(4,114,83,178),(5,113,84,177),(6,128,85,192),(7,127,86,191),(8,126,87,190),(9,125,88,189),(10,124,89,188),(11,123,90,187),(12,122,91,186),(13,121,92,185),(14,120,93,184),(15,119,94,183),(16,118,95,182),(17,108,42,151),(18,107,43,150),(19,106,44,149),(20,105,45,148),(21,104,46,147),(22,103,47,146),(23,102,48,145),(24,101,33,160),(25,100,34,159),(26,99,35,158),(27,98,36,157),(28,97,37,156),(29,112,38,155),(30,111,39,154),(31,110,40,153),(32,109,41,152),(49,71,136,173),(50,70,137,172),(51,69,138,171),(52,68,139,170),(53,67,140,169),(54,66,141,168),(55,65,142,167),(56,80,143,166),(57,79,144,165),(58,78,129,164),(59,77,130,163),(60,76,131,162),(61,75,132,161),(62,74,133,176),(63,73,134,175),(64,72,135,174)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | Q16 | D8 | Dic6 | C4×S3 | C3⋊D4 | D16 | Q32 | C3⋊Q16 | D4⋊S3 | C3⋊D16 | C3⋊Q32 |
| kernel | C6.6D16 | C2×C3⋊C16 | C24⋊1C4 | C3×C2.D8 | C3⋊C16 | C2.D8 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C6.6D16 ►in GL6(𝔽97)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 |
| 0 | 0 | 0 | 0 | 1 | 96 |
| 24 | 45 | 0 | 0 | 0 | 0 |
| 26 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 2 | 0 | 0 |
| 0 | 0 | 95 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 80 |
| 0 | 0 | 0 | 0 | 33 | 47 |
| 42 | 64 | 0 | 0 | 0 | 0 |
| 74 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 31 | 0 | 0 |
| 0 | 0 | 31 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 74 | 89 |
| 0 | 0 | 0 | 0 | 66 | 23 |
G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[24,26,0,0,0,0,45,69,0,0,0,0,0,0,26,95,0,0,0,0,2,26,0,0,0,0,0,0,50,33,0,0,0,0,80,47],[42,74,0,0,0,0,64,55,0,0,0,0,0,0,69,31,0,0,0,0,31,28,0,0,0,0,0,0,74,66,0,0,0,0,89,23] >;
C6.6D16 in GAP, Magma, Sage, TeX
C_6._6D_{16} % in TeX
G:=Group("C6.6D16"); // GroupNames label
G:=SmallGroup(192,48);
// by ID
G=gap.SmallGroup(192,48);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,346,192,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^6=b^16=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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