metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C51⋊3D4, C17⋊2D12, D6⋊2D17, Dic17⋊S3, C6.6D34, C34.6D6, D102⋊4C2, C102.6C22, (S3×C34)⋊2C2, C3⋊1(C17⋊D4), C2.6(S3×D17), (C3×Dic17)⋊3C2, SmallGroup(408,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17⋊D12
G = < a,b,c | a17=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C17⋊D12
►On 204 points
Generators in S204
(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)(188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 145 87 159 31 122 55 197 49 111 84 172)(2 144 88 158 32 121 56 196 50 110 85 171)(3 143 89 157 33 120 57 195 51 109 69 187)(4 142 90 156 34 136 58 194 35 108 70 186)(5 141 91 155 18 135 59 193 36 107 71 185)(6 140 92 154 19 134 60 192 37 106 72 184)(7 139 93 170 20 133 61 191 38 105 73 183)(8 138 94 169 21 132 62 190 39 104 74 182)(9 137 95 168 22 131 63 189 40 103 75 181)(10 153 96 167 23 130 64 188 41 119 76 180)(11 152 97 166 24 129 65 204 42 118 77 179)(12 151 98 165 25 128 66 203 43 117 78 178)(13 150 99 164 26 127 67 202 44 116 79 177)(14 149 100 163 27 126 68 201 45 115 80 176)(15 148 101 162 28 125 52 200 46 114 81 175)(16 147 102 161 29 124 53 199 47 113 82 174)(17 146 86 160 30 123 54 198 48 112 83 173)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 51)(17 50)(18 27)(19 26)(20 25)(21 24)(22 23)(28 34)(29 33)(30 32)(52 90)(53 89)(54 88)(55 87)(56 86)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(83 85)(103 180)(104 179)(105 178)(106 177)(107 176)(108 175)(109 174)(110 173)(111 172)(112 171)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 161)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 170)(129 169)(130 168)(131 167)(132 166)(133 165)(134 164)(135 163)(136 162)(137 188)(138 204)(139 203)(140 202)(141 201)(142 200)(143 199)(144 198)(145 197)(146 196)(147 195)(148 194)(149 193)(150 192)(151 191)(152 190)(153 189)
G:=sub<Sym(204)| (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)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,145,87,159,31,122,55,197,49,111,84,172)(2,144,88,158,32,121,56,196,50,110,85,171)(3,143,89,157,33,120,57,195,51,109,69,187)(4,142,90,156,34,136,58,194,35,108,70,186)(5,141,91,155,18,135,59,193,36,107,71,185)(6,140,92,154,19,134,60,192,37,106,72,184)(7,139,93,170,20,133,61,191,38,105,73,183)(8,138,94,169,21,132,62,190,39,104,74,182)(9,137,95,168,22,131,63,189,40,103,75,181)(10,153,96,167,23,130,64,188,41,119,76,180)(11,152,97,166,24,129,65,204,42,118,77,179)(12,151,98,165,25,128,66,203,43,117,78,178)(13,150,99,164,26,127,67,202,44,116,79,177)(14,149,100,163,27,126,68,201,45,115,80,176)(15,148,101,162,28,125,52,200,46,114,81,175)(16,147,102,161,29,124,53,199,47,113,82,174)(17,146,86,160,30,123,54,198,48,112,83,173), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,51)(17,50)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(52,90)(53,89)(54,88)(55,87)(56,86)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(83,85)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175)(109,174)(110,173)(111,172)(112,171)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,188)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189)>;
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)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,145,87,159,31,122,55,197,49,111,84,172)(2,144,88,158,32,121,56,196,50,110,85,171)(3,143,89,157,33,120,57,195,51,109,69,187)(4,142,90,156,34,136,58,194,35,108,70,186)(5,141,91,155,18,135,59,193,36,107,71,185)(6,140,92,154,19,134,60,192,37,106,72,184)(7,139,93,170,20,133,61,191,38,105,73,183)(8,138,94,169,21,132,62,190,39,104,74,182)(9,137,95,168,22,131,63,189,40,103,75,181)(10,153,96,167,23,130,64,188,41,119,76,180)(11,152,97,166,24,129,65,204,42,118,77,179)(12,151,98,165,25,128,66,203,43,117,78,178)(13,150,99,164,26,127,67,202,44,116,79,177)(14,149,100,163,27,126,68,201,45,115,80,176)(15,148,101,162,28,125,52,200,46,114,81,175)(16,147,102,161,29,124,53,199,47,113,82,174)(17,146,86,160,30,123,54,198,48,112,83,173), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,51)(17,50)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(52,90)(53,89)(54,88)(55,87)(56,86)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(83,85)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175)(109,174)(110,173)(111,172)(112,171)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,188)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189) );
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),(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,145,87,159,31,122,55,197,49,111,84,172),(2,144,88,158,32,121,56,196,50,110,85,171),(3,143,89,157,33,120,57,195,51,109,69,187),(4,142,90,156,34,136,58,194,35,108,70,186),(5,141,91,155,18,135,59,193,36,107,71,185),(6,140,92,154,19,134,60,192,37,106,72,184),(7,139,93,170,20,133,61,191,38,105,73,183),(8,138,94,169,21,132,62,190,39,104,74,182),(9,137,95,168,22,131,63,189,40,103,75,181),(10,153,96,167,23,130,64,188,41,119,76,180),(11,152,97,166,24,129,65,204,42,118,77,179),(12,151,98,165,25,128,66,203,43,117,78,178),(13,150,99,164,26,127,67,202,44,116,79,177),(14,149,100,163,27,126,68,201,45,115,80,176),(15,148,101,162,28,125,52,200,46,114,81,175),(16,147,102,161,29,124,53,199,47,113,82,174),(17,146,86,160,30,123,54,198,48,112,83,173)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,51),(17,50),(18,27),(19,26),(20,25),(21,24),(22,23),(28,34),(29,33),(30,32),(52,90),(53,89),(54,88),(55,87),(56,86),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(83,85),(103,180),(104,179),(105,178),(106,177),(107,176),(108,175),(109,174),(110,173),(111,172),(112,171),(113,187),(114,186),(115,185),(116,184),(117,183),(118,182),(119,181),(120,161),(121,160),(122,159),(123,158),(124,157),(125,156),(126,155),(127,154),(128,170),(129,169),(130,168),(131,167),(132,166),(133,165),(134,164),(135,163),(136,162),(137,188),(138,204),(139,203),(140,202),(141,201),(142,200),(143,199),(144,198),(145,197),(146,196),(147,195),(148,194),(149,193),(150,192),(151,191),(152,190),(153,189)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 51A | ··· | 51H | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 6 | 102 | 2 | 34 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D12 | D17 | D34 | C17⋊D4 | S3×D17 | C17⋊D12 |
| kernel | C17⋊D12 | C3×Dic17 | S3×C34 | D102 | Dic17 | C51 | C34 | C17 | D6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 | 8 |
Matrix representation of C17⋊D12 ►in GL4(𝔽409) generated by
| 83 | 1 | 0 | 0 |
| 20 | 222 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 19 | 95 | 0 | 0 |
| 332 | 390 | 0 | 0 |
| 0 | 0 | 1 | 85 |
| 0 | 0 | 178 | 407 |
| 168 | 2 | 0 | 0 |
| 408 | 241 | 0 | 0 |
| 0 | 0 | 2 | 85 |
| 0 | 0 | 178 | 407 |
G:=sub<GL(4,GF(409))| [83,20,0,0,1,222,0,0,0,0,1,0,0,0,0,1],[19,332,0,0,95,390,0,0,0,0,1,178,0,0,85,407],[168,408,0,0,2,241,0,0,0,0,2,178,0,0,85,407] >;
C17⋊D12 in GAP, Magma, Sage, TeX
C_{17}\rtimes D_{12} % in TeX
G:=Group("C17:D12"); // GroupNames label
G:=SmallGroup(408,12);
// by ID
G=gap.SmallGroup(408,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,61,168,9604]);
// Polycyclic
G:=Group<a,b,c|a^17=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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