metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.17D8, C12.8Q16, C42.220D6, C3⋊C8⋊7Q8, C4⋊Q8.4S3, C4⋊C4.79D6, C3⋊3(C8⋊2Q8), C6.59(C2×D8), C4.34(S3×Q8), C4.6(D4⋊S3), C6.30(C4⋊Q8), C6.40(C2×Q16), C12.35(C2×Q8), (C2×C12).151D4, C4.3(C3⋊Q16), C12⋊2Q8.18C2, C6.Q16.15C2, (C2×C12).396C23, (C4×C12).125C22, C4⋊Dic3.156C22, C2.10(Dic3⋊Q8), (C4×C3⋊C8).11C2, (C3×C4⋊Q8).4C2, C2.14(C2×D4⋊S3), (C2×C6).527(C2×D4), C2.11(C2×C3⋊Q16), (C2×C3⋊C8).259C22, (C2×C4).133(C3⋊D4), (C3×C4⋊C4).126C22, (C2×C4).493(C22×S3), C22.199(C2×C3⋊D4), SmallGroup(192,637)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.17D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=b-1 >
Subgroups: 240 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, C2.D8, C4⋊Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C8⋊2Q8, C4×C3⋊C8, C6.Q16, C12⋊2Q8, C3×C4⋊Q8, C12.17D8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, Q16, C2×D4, C2×Q8, C3⋊D4, C22×S3, C4⋊Q8, C2×D8, C2×Q16, D4⋊S3, C3⋊Q16, S3×Q8, C2×C3⋊D4, C8⋊2Q8, C2×D4⋊S3, C2×C3⋊Q16, Dic3⋊Q8, C12.17D8
►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 86 42 69 144 149 169 182)(2 91 43 62 133 154 170 187)(3 96 44 67 134 147 171 192)(4 89 45 72 135 152 172 185)(5 94 46 65 136 145 173 190)(6 87 47 70 137 150 174 183)(7 92 48 63 138 155 175 188)(8 85 37 68 139 148 176 181)(9 90 38 61 140 153 177 186)(10 95 39 66 141 146 178 191)(11 88 40 71 142 151 179 184)(12 93 41 64 143 156 180 189)(13 159 58 83 29 100 127 114)(14 164 59 76 30 105 128 119)(15 157 60 81 31 98 129 112)(16 162 49 74 32 103 130 117)(17 167 50 79 33 108 131 110)(18 160 51 84 34 101 132 115)(19 165 52 77 35 106 121 120)(20 158 53 82 36 99 122 113)(21 163 54 75 25 104 123 118)(22 168 55 80 26 97 124 111)(23 161 56 73 27 102 125 116)(24 166 57 78 28 107 126 109)
(1 131 7 125)(2 130 8 124)(3 129 9 123)(4 128 10 122)(5 127 11 121)(6 126 12 132)(13 179 19 173)(14 178 20 172)(15 177 21 171)(16 176 22 170)(17 175 23 169)(18 174 24 180)(25 44 31 38)(26 43 32 37)(27 42 33 48)(28 41 34 47)(29 40 35 46)(30 39 36 45)(49 139 55 133)(50 138 56 144)(51 137 57 143)(52 136 58 142)(53 135 59 141)(54 134 60 140)(61 75 67 81)(62 74 68 80)(63 73 69 79)(64 84 70 78)(65 83 71 77)(66 82 72 76)(85 97 91 103)(86 108 92 102)(87 107 93 101)(88 106 94 100)(89 105 95 99)(90 104 96 98)(109 189 115 183)(110 188 116 182)(111 187 117 181)(112 186 118 192)(113 185 119 191)(114 184 120 190)(145 159 151 165)(146 158 152 164)(147 157 153 163)(148 168 154 162)(149 167 155 161)(150 166 156 160)
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), (1,86,42,69,144,149,169,182)(2,91,43,62,133,154,170,187)(3,96,44,67,134,147,171,192)(4,89,45,72,135,152,172,185)(5,94,46,65,136,145,173,190)(6,87,47,70,137,150,174,183)(7,92,48,63,138,155,175,188)(8,85,37,68,139,148,176,181)(9,90,38,61,140,153,177,186)(10,95,39,66,141,146,178,191)(11,88,40,71,142,151,179,184)(12,93,41,64,143,156,180,189)(13,159,58,83,29,100,127,114)(14,164,59,76,30,105,128,119)(15,157,60,81,31,98,129,112)(16,162,49,74,32,103,130,117)(17,167,50,79,33,108,131,110)(18,160,51,84,34,101,132,115)(19,165,52,77,35,106,121,120)(20,158,53,82,36,99,122,113)(21,163,54,75,25,104,123,118)(22,168,55,80,26,97,124,111)(23,161,56,73,27,102,125,116)(24,166,57,78,28,107,126,109), (1,131,7,125)(2,130,8,124)(3,129,9,123)(4,128,10,122)(5,127,11,121)(6,126,12,132)(13,179,19,173)(14,178,20,172)(15,177,21,171)(16,176,22,170)(17,175,23,169)(18,174,24,180)(25,44,31,38)(26,43,32,37)(27,42,33,48)(28,41,34,47)(29,40,35,46)(30,39,36,45)(49,139,55,133)(50,138,56,144)(51,137,57,143)(52,136,58,142)(53,135,59,141)(54,134,60,140)(61,75,67,81)(62,74,68,80)(63,73,69,79)(64,84,70,78)(65,83,71,77)(66,82,72,76)(85,97,91,103)(86,108,92,102)(87,107,93,101)(88,106,94,100)(89,105,95,99)(90,104,96,98)(109,189,115,183)(110,188,116,182)(111,187,117,181)(112,186,118,192)(113,185,119,191)(114,184,120,190)(145,159,151,165)(146,158,152,164)(147,157,153,163)(148,168,154,162)(149,167,155,161)(150,166,156,160)>;
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), (1,86,42,69,144,149,169,182)(2,91,43,62,133,154,170,187)(3,96,44,67,134,147,171,192)(4,89,45,72,135,152,172,185)(5,94,46,65,136,145,173,190)(6,87,47,70,137,150,174,183)(7,92,48,63,138,155,175,188)(8,85,37,68,139,148,176,181)(9,90,38,61,140,153,177,186)(10,95,39,66,141,146,178,191)(11,88,40,71,142,151,179,184)(12,93,41,64,143,156,180,189)(13,159,58,83,29,100,127,114)(14,164,59,76,30,105,128,119)(15,157,60,81,31,98,129,112)(16,162,49,74,32,103,130,117)(17,167,50,79,33,108,131,110)(18,160,51,84,34,101,132,115)(19,165,52,77,35,106,121,120)(20,158,53,82,36,99,122,113)(21,163,54,75,25,104,123,118)(22,168,55,80,26,97,124,111)(23,161,56,73,27,102,125,116)(24,166,57,78,28,107,126,109), (1,131,7,125)(2,130,8,124)(3,129,9,123)(4,128,10,122)(5,127,11,121)(6,126,12,132)(13,179,19,173)(14,178,20,172)(15,177,21,171)(16,176,22,170)(17,175,23,169)(18,174,24,180)(25,44,31,38)(26,43,32,37)(27,42,33,48)(28,41,34,47)(29,40,35,46)(30,39,36,45)(49,139,55,133)(50,138,56,144)(51,137,57,143)(52,136,58,142)(53,135,59,141)(54,134,60,140)(61,75,67,81)(62,74,68,80)(63,73,69,79)(64,84,70,78)(65,83,71,77)(66,82,72,76)(85,97,91,103)(86,108,92,102)(87,107,93,101)(88,106,94,100)(89,105,95,99)(90,104,96,98)(109,189,115,183)(110,188,116,182)(111,187,117,181)(112,186,118,192)(113,185,119,191)(114,184,120,190)(145,159,151,165)(146,158,152,164)(147,157,153,163)(148,168,154,162)(149,167,155,161)(150,166,156,160) );
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)], [(1,86,42,69,144,149,169,182),(2,91,43,62,133,154,170,187),(3,96,44,67,134,147,171,192),(4,89,45,72,135,152,172,185),(5,94,46,65,136,145,173,190),(6,87,47,70,137,150,174,183),(7,92,48,63,138,155,175,188),(8,85,37,68,139,148,176,181),(9,90,38,61,140,153,177,186),(10,95,39,66,141,146,178,191),(11,88,40,71,142,151,179,184),(12,93,41,64,143,156,180,189),(13,159,58,83,29,100,127,114),(14,164,59,76,30,105,128,119),(15,157,60,81,31,98,129,112),(16,162,49,74,32,103,130,117),(17,167,50,79,33,108,131,110),(18,160,51,84,34,101,132,115),(19,165,52,77,35,106,121,120),(20,158,53,82,36,99,122,113),(21,163,54,75,25,104,123,118),(22,168,55,80,26,97,124,111),(23,161,56,73,27,102,125,116),(24,166,57,78,28,107,126,109)], [(1,131,7,125),(2,130,8,124),(3,129,9,123),(4,128,10,122),(5,127,11,121),(6,126,12,132),(13,179,19,173),(14,178,20,172),(15,177,21,171),(16,176,22,170),(17,175,23,169),(18,174,24,180),(25,44,31,38),(26,43,32,37),(27,42,33,48),(28,41,34,47),(29,40,35,46),(30,39,36,45),(49,139,55,133),(50,138,56,144),(51,137,57,143),(52,136,58,142),(53,135,59,141),(54,134,60,140),(61,75,67,81),(62,74,68,80),(63,73,69,79),(64,84,70,78),(65,83,71,77),(66,82,72,76),(85,97,91,103),(86,108,92,102),(87,107,93,101),(88,106,94,100),(89,105,95,99),(90,104,96,98),(109,189,115,183),(110,188,116,182),(111,187,117,181),(112,186,118,192),(113,185,119,191),(114,184,120,190),(145,159,151,165),(146,158,152,164),(147,157,153,163),(148,168,154,162),(149,167,155,161),(150,166,156,160)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 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 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | D8 | Q16 | C3⋊D4 | D4⋊S3 | C3⋊Q16 | S3×Q8 |
| kernel | C12.17D8 | C4×C3⋊C8 | C6.Q16 | C12⋊2Q8 | C3×C4⋊Q8 | C4⋊Q8 | C3⋊C8 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C12.17D8 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 57 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 14 | 0 | 0 |
| 0 | 0 | 68 | 54 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 19 | 52 | 0 | 0 | 0 | 0 |
| 52 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 10 | 0 | 0 |
| 0 | 0 | 32 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 14 |
| 0 | 0 | 0 | 0 | 14 | 30 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,57,0,0,0,0,16,16,0,0,0,0,0,0,19,68,0,0,0,0,14,54,0,0,0,0,0,0,16,57,0,0,0,0,16,16],[19,52,0,0,0,0,52,54,0,0,0,0,0,0,51,32,0,0,0,0,10,22,0,0,0,0,0,0,43,14,0,0,0,0,14,30] >;
C12.17D8 in GAP, Magma, Sage, TeX
C_{12}._{17}D_8 % in TeX
G:=Group("C12.17D8"); // GroupNames label
G:=SmallGroup(192,637);
// by ID
G=gap.SmallGroup(192,637);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,422,135,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations