metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.5Q16, C12.9SD16, C42.10D6, C4⋊Q8.1S3, (C6×Q8).2C4, (C2×C12).107D4, (C2×Q8).5Dic3, C12⋊C8.12C2, C4.6(C3⋊Q16), C3⋊2(C4.6Q16), (C4×C12).48C22, C4.8(Q8⋊2S3), C6.12(Q8⋊C4), C6.10(C4.D4), C2.5(C12.D4), C2.4(Q8⋊2Dic3), C22.42(C6.D4), (C3×C4⋊Q8).1C2, (C2×C12).172(C2×C4), (C2×C4).12(C2×Dic3), (C2×C4).177(C3⋊D4), (C2×C6).104(C22⋊C4), SmallGroup(192,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.5Q16
G = < a,b,c | a12=b8=1, c2=a6b4, bab-1=a-1, cac-1=a7, cbc-1=a3b-1 >
Subgroups: 144 in 64 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×Q8, C4⋊C8, C4⋊Q8, C2×C3⋊C8, C4×C12, C3×C4⋊C4, C6×Q8, C4.6Q16, C12⋊C8, C3×C4⋊Q8, C12.5Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, C2×Dic3, C3⋊D4, C4.D4, Q8⋊C4, Q8⋊2S3, C3⋊Q16, C6.D4, C4.6Q16, C12.D4, Q8⋊2Dic3, C12.5Q16
►Regular action on 192 points
Generators in S192
(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)
(1 182 166 121 59 156 47 34)(2 181 167 132 60 155 48 33)(3 192 168 131 49 154 37 32)(4 191 157 130 50 153 38 31)(5 190 158 129 51 152 39 30)(6 189 159 128 52 151 40 29)(7 188 160 127 53 150 41 28)(8 187 161 126 54 149 42 27)(9 186 162 125 55 148 43 26)(10 185 163 124 56 147 44 25)(11 184 164 123 57 146 45 36)(12 183 165 122 58 145 46 35)(13 85 97 79 143 66 113 177)(14 96 98 78 144 65 114 176)(15 95 99 77 133 64 115 175)(16 94 100 76 134 63 116 174)(17 93 101 75 135 62 117 173)(18 92 102 74 136 61 118 172)(19 91 103 73 137 72 119 171)(20 90 104 84 138 71 120 170)(21 89 105 83 139 70 109 169)(22 88 106 82 140 69 110 180)(23 87 107 81 141 68 111 179)(24 86 108 80 142 67 112 178)
(1 77 53 169)(2 84 54 176)(3 79 55 171)(4 74 56 178)(5 81 57 173)(6 76 58 180)(7 83 59 175)(8 78 60 170)(9 73 49 177)(10 80 50 172)(11 75 51 179)(12 82 52 174)(13 35 137 128)(14 30 138 123)(15 25 139 130)(16 32 140 125)(17 27 141 132)(18 34 142 127)(19 29 143 122)(20 36 144 129)(21 31 133 124)(22 26 134 131)(23 33 135 126)(24 28 136 121)(37 66 162 91)(38 61 163 86)(39 68 164 93)(40 63 165 88)(41 70 166 95)(42 65 167 90)(43 72 168 85)(44 67 157 92)(45 62 158 87)(46 69 159 94)(47 64 160 89)(48 71 161 96)(97 145 119 189)(98 152 120 184)(99 147 109 191)(100 154 110 186)(101 149 111 181)(102 156 112 188)(103 151 113 183)(104 146 114 190)(105 153 115 185)(106 148 116 192)(107 155 117 187)(108 150 118 182)
G:=sub<Sym(192)| (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), (1,182,166,121,59,156,47,34)(2,181,167,132,60,155,48,33)(3,192,168,131,49,154,37,32)(4,191,157,130,50,153,38,31)(5,190,158,129,51,152,39,30)(6,189,159,128,52,151,40,29)(7,188,160,127,53,150,41,28)(8,187,161,126,54,149,42,27)(9,186,162,125,55,148,43,26)(10,185,163,124,56,147,44,25)(11,184,164,123,57,146,45,36)(12,183,165,122,58,145,46,35)(13,85,97,79,143,66,113,177)(14,96,98,78,144,65,114,176)(15,95,99,77,133,64,115,175)(16,94,100,76,134,63,116,174)(17,93,101,75,135,62,117,173)(18,92,102,74,136,61,118,172)(19,91,103,73,137,72,119,171)(20,90,104,84,138,71,120,170)(21,89,105,83,139,70,109,169)(22,88,106,82,140,69,110,180)(23,87,107,81,141,68,111,179)(24,86,108,80,142,67,112,178), (1,77,53,169)(2,84,54,176)(3,79,55,171)(4,74,56,178)(5,81,57,173)(6,76,58,180)(7,83,59,175)(8,78,60,170)(9,73,49,177)(10,80,50,172)(11,75,51,179)(12,82,52,174)(13,35,137,128)(14,30,138,123)(15,25,139,130)(16,32,140,125)(17,27,141,132)(18,34,142,127)(19,29,143,122)(20,36,144,129)(21,31,133,124)(22,26,134,131)(23,33,135,126)(24,28,136,121)(37,66,162,91)(38,61,163,86)(39,68,164,93)(40,63,165,88)(41,70,166,95)(42,65,167,90)(43,72,168,85)(44,67,157,92)(45,62,158,87)(46,69,159,94)(47,64,160,89)(48,71,161,96)(97,145,119,189)(98,152,120,184)(99,147,109,191)(100,154,110,186)(101,149,111,181)(102,156,112,188)(103,151,113,183)(104,146,114,190)(105,153,115,185)(106,148,116,192)(107,155,117,187)(108,150,118,182)>;
G:=Group( (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), (1,182,166,121,59,156,47,34)(2,181,167,132,60,155,48,33)(3,192,168,131,49,154,37,32)(4,191,157,130,50,153,38,31)(5,190,158,129,51,152,39,30)(6,189,159,128,52,151,40,29)(7,188,160,127,53,150,41,28)(8,187,161,126,54,149,42,27)(9,186,162,125,55,148,43,26)(10,185,163,124,56,147,44,25)(11,184,164,123,57,146,45,36)(12,183,165,122,58,145,46,35)(13,85,97,79,143,66,113,177)(14,96,98,78,144,65,114,176)(15,95,99,77,133,64,115,175)(16,94,100,76,134,63,116,174)(17,93,101,75,135,62,117,173)(18,92,102,74,136,61,118,172)(19,91,103,73,137,72,119,171)(20,90,104,84,138,71,120,170)(21,89,105,83,139,70,109,169)(22,88,106,82,140,69,110,180)(23,87,107,81,141,68,111,179)(24,86,108,80,142,67,112,178), (1,77,53,169)(2,84,54,176)(3,79,55,171)(4,74,56,178)(5,81,57,173)(6,76,58,180)(7,83,59,175)(8,78,60,170)(9,73,49,177)(10,80,50,172)(11,75,51,179)(12,82,52,174)(13,35,137,128)(14,30,138,123)(15,25,139,130)(16,32,140,125)(17,27,141,132)(18,34,142,127)(19,29,143,122)(20,36,144,129)(21,31,133,124)(22,26,134,131)(23,33,135,126)(24,28,136,121)(37,66,162,91)(38,61,163,86)(39,68,164,93)(40,63,165,88)(41,70,166,95)(42,65,167,90)(43,72,168,85)(44,67,157,92)(45,62,158,87)(46,69,159,94)(47,64,160,89)(48,71,161,96)(97,145,119,189)(98,152,120,184)(99,147,109,191)(100,154,110,186)(101,149,111,181)(102,156,112,188)(103,151,113,183)(104,146,114,190)(105,153,115,185)(106,148,116,192)(107,155,117,187)(108,150,118,182) );
G=PermutationGroup([[(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)], [(1,182,166,121,59,156,47,34),(2,181,167,132,60,155,48,33),(3,192,168,131,49,154,37,32),(4,191,157,130,50,153,38,31),(5,190,158,129,51,152,39,30),(6,189,159,128,52,151,40,29),(7,188,160,127,53,150,41,28),(8,187,161,126,54,149,42,27),(9,186,162,125,55,148,43,26),(10,185,163,124,56,147,44,25),(11,184,164,123,57,146,45,36),(12,183,165,122,58,145,46,35),(13,85,97,79,143,66,113,177),(14,96,98,78,144,65,114,176),(15,95,99,77,133,64,115,175),(16,94,100,76,134,63,116,174),(17,93,101,75,135,62,117,173),(18,92,102,74,136,61,118,172),(19,91,103,73,137,72,119,171),(20,90,104,84,138,71,120,170),(21,89,105,83,139,70,109,169),(22,88,106,82,140,69,110,180),(23,87,107,81,141,68,111,179),(24,86,108,80,142,67,112,178)], [(1,77,53,169),(2,84,54,176),(3,79,55,171),(4,74,56,178),(5,81,57,173),(6,76,58,180),(7,83,59,175),(8,78,60,170),(9,73,49,177),(10,80,50,172),(11,75,51,179),(12,82,52,174),(13,35,137,128),(14,30,138,123),(15,25,139,130),(16,32,140,125),(17,27,141,132),(18,34,142,127),(19,29,143,122),(20,36,144,129),(21,31,133,124),(22,26,134,131),(23,33,135,126),(24,28,136,121),(37,66,162,91),(38,61,163,86),(39,68,164,93),(40,63,165,88),(41,70,166,95),(42,65,167,90),(43,72,168,85),(44,67,157,92),(45,62,158,87),(46,69,159,94),(47,64,160,89),(48,71,161,96),(97,145,119,189),(98,152,120,184),(99,147,109,191),(100,154,110,186),(101,149,111,181),(102,156,112,188),(103,151,113,183),(104,146,114,190),(105,153,115,185),(106,148,116,192),(107,155,117,187),(108,150,118,182)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | SD16 | Q16 | C3⋊D4 | C4.D4 | Q8⋊2S3 | C3⋊Q16 | C12.D4 |
| kernel | C12.5Q16 | C12⋊C8 | C3×C4⋊Q8 | C6×Q8 | C4⋊Q8 | C2×C12 | C42 | C2×Q8 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C12.5Q16 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 49 | 70 | 0 | 0 | 0 | 0 |
| 70 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 39 |
| 0 | 0 | 0 | 0 | 65 | 47 |
| 53 | 16 | 0 | 0 | 0 | 0 |
| 16 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 21 | 0 | 0 |
| 0 | 0 | 21 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[49,70,0,0,0,0,70,24,0,0,0,0,0,0,6,67,0,0,0,0,6,6,0,0,0,0,0,0,26,65,0,0,0,0,39,47],[53,16,0,0,0,0,16,20,0,0,0,0,0,0,54,21,0,0,0,0,21,19,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;
C12.5Q16 in GAP, Magma, Sage, TeX
C_{12}._5Q_{16} % in TeX
G:=Group("C12.5Q16"); // GroupNames label
G:=SmallGroup(192,105);
// by ID
G=gap.SmallGroup(192,105);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,100,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b^-1>;
// generators/relations