metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C51⋊7D4, D102⋊2C2, C6.12D34, C34.12D6, C2.5D102, C22⋊2D51, Dic51⋊1C2, C102.12C22, (C2×C34)⋊4S3, (C2×C6)⋊2D17, (C2×C102)⋊2C2, C3⋊3(C17⋊D4), C17⋊3(C3⋊D4), SmallGroup(408,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C51⋊7D4
G = < a,b,c | a51=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C51⋊7D4
►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)
(1 104 61 198)(2 103 62 197)(3 153 63 196)(4 152 64 195)(5 151 65 194)(6 150 66 193)(7 149 67 192)(8 148 68 191)(9 147 69 190)(10 146 70 189)(11 145 71 188)(12 144 72 187)(13 143 73 186)(14 142 74 185)(15 141 75 184)(16 140 76 183)(17 139 77 182)(18 138 78 181)(19 137 79 180)(20 136 80 179)(21 135 81 178)(22 134 82 177)(23 133 83 176)(24 132 84 175)(25 131 85 174)(26 130 86 173)(27 129 87 172)(28 128 88 171)(29 127 89 170)(30 126 90 169)(31 125 91 168)(32 124 92 167)(33 123 93 166)(34 122 94 165)(35 121 95 164)(36 120 96 163)(37 119 97 162)(38 118 98 161)(39 117 99 160)(40 116 100 159)(41 115 101 158)(42 114 102 157)(43 113 52 156)(44 112 53 155)(45 111 54 154)(46 110 55 204)(47 109 56 203)(48 108 57 202)(49 107 58 201)(50 106 59 200)(51 105 60 199)
(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(103 199)(104 198)(105 197)(106 196)(107 195)(108 194)(109 193)(110 192)(111 191)(112 190)(113 189)(114 188)(115 187)(116 186)(117 185)(118 184)(119 183)(120 182)(121 181)(122 180)(123 179)(124 178)(125 177)(126 176)(127 175)(128 174)(129 173)(130 172)(131 171)(132 170)(133 169)(134 168)(135 167)(136 166)(137 165)(138 164)(139 163)(140 162)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 204)(150 203)(151 202)(152 201)(153 200)
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), (1,104,61,198)(2,103,62,197)(3,153,63,196)(4,152,64,195)(5,151,65,194)(6,150,66,193)(7,149,67,192)(8,148,68,191)(9,147,69,190)(10,146,70,189)(11,145,71,188)(12,144,72,187)(13,143,73,186)(14,142,74,185)(15,141,75,184)(16,140,76,183)(17,139,77,182)(18,138,78,181)(19,137,79,180)(20,136,80,179)(21,135,81,178)(22,134,82,177)(23,133,83,176)(24,132,84,175)(25,131,85,174)(26,130,86,173)(27,129,87,172)(28,128,88,171)(29,127,89,170)(30,126,90,169)(31,125,91,168)(32,124,92,167)(33,123,93,166)(34,122,94,165)(35,121,95,164)(36,120,96,163)(37,119,97,162)(38,118,98,161)(39,117,99,160)(40,116,100,159)(41,115,101,158)(42,114,102,157)(43,113,52,156)(44,112,53,155)(45,111,54,154)(46,110,55,204)(47,109,56,203)(48,108,57,202)(49,107,58,201)(50,106,59,200)(51,105,60,199), (2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(103,199)(104,198)(105,197)(106,196)(107,195)(108,194)(109,193)(110,192)(111,191)(112,190)(113,189)(114,188)(115,187)(116,186)(117,185)(118,184)(119,183)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,175)(128,174)(129,173)(130,172)(131,171)(132,170)(133,169)(134,168)(135,167)(136,166)(137,165)(138,164)(139,163)(140,162)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,204)(150,203)(151,202)(152,201)(153,200)>;
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), (1,104,61,198)(2,103,62,197)(3,153,63,196)(4,152,64,195)(5,151,65,194)(6,150,66,193)(7,149,67,192)(8,148,68,191)(9,147,69,190)(10,146,70,189)(11,145,71,188)(12,144,72,187)(13,143,73,186)(14,142,74,185)(15,141,75,184)(16,140,76,183)(17,139,77,182)(18,138,78,181)(19,137,79,180)(20,136,80,179)(21,135,81,178)(22,134,82,177)(23,133,83,176)(24,132,84,175)(25,131,85,174)(26,130,86,173)(27,129,87,172)(28,128,88,171)(29,127,89,170)(30,126,90,169)(31,125,91,168)(32,124,92,167)(33,123,93,166)(34,122,94,165)(35,121,95,164)(36,120,96,163)(37,119,97,162)(38,118,98,161)(39,117,99,160)(40,116,100,159)(41,115,101,158)(42,114,102,157)(43,113,52,156)(44,112,53,155)(45,111,54,154)(46,110,55,204)(47,109,56,203)(48,108,57,202)(49,107,58,201)(50,106,59,200)(51,105,60,199), (2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(103,199)(104,198)(105,197)(106,196)(107,195)(108,194)(109,193)(110,192)(111,191)(112,190)(113,189)(114,188)(115,187)(116,186)(117,185)(118,184)(119,183)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,175)(128,174)(129,173)(130,172)(131,171)(132,170)(133,169)(134,168)(135,167)(136,166)(137,165)(138,164)(139,163)(140,162)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,204)(150,203)(151,202)(152,201)(153,200) );
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)], [(1,104,61,198),(2,103,62,197),(3,153,63,196),(4,152,64,195),(5,151,65,194),(6,150,66,193),(7,149,67,192),(8,148,68,191),(9,147,69,190),(10,146,70,189),(11,145,71,188),(12,144,72,187),(13,143,73,186),(14,142,74,185),(15,141,75,184),(16,140,76,183),(17,139,77,182),(18,138,78,181),(19,137,79,180),(20,136,80,179),(21,135,81,178),(22,134,82,177),(23,133,83,176),(24,132,84,175),(25,131,85,174),(26,130,86,173),(27,129,87,172),(28,128,88,171),(29,127,89,170),(30,126,90,169),(31,125,91,168),(32,124,92,167),(33,123,93,166),(34,122,94,165),(35,121,95,164),(36,120,96,163),(37,119,97,162),(38,118,98,161),(39,117,99,160),(40,116,100,159),(41,115,101,158),(42,114,102,157),(43,113,52,156),(44,112,53,155),(45,111,54,154),(46,110,55,204),(47,109,56,203),(48,108,57,202),(49,107,58,201),(50,106,59,200),(51,105,60,199)], [(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(103,199),(104,198),(105,197),(106,196),(107,195),(108,194),(109,193),(110,192),(111,191),(112,190),(113,189),(114,188),(115,187),(116,186),(117,185),(118,184),(119,183),(120,182),(121,181),(122,180),(123,179),(124,178),(125,177),(126,176),(127,175),(128,174),(129,173),(130,172),(131,171),(132,170),(133,169),(134,168),(135,167),(136,166),(137,165),(138,164),(139,163),(140,162),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,204),(150,203),(151,202),(152,201),(153,200)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 17A | ··· | 17H | 34A | ··· | 34X | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 102 | 2 | 102 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D17 | D34 | D51 | C17⋊D4 | D102 | C51⋊7D4 |
| kernel | C51⋊7D4 | Dic51 | D102 | C2×C102 | C2×C34 | C51 | C34 | C17 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C51⋊7D4 ►in GL2(𝔽409) generated by
| 221 | 157 |
| 224 | 198 |
| 356 | 216 |
| 218 | 53 |
| 146 | 94 |
| 204 | 263 |
G:=sub<GL(2,GF(409))| [221,224,157,198],[356,218,216,53],[146,204,94,263] >;
C51⋊7D4 in GAP, Magma, Sage, TeX
C_{51}\rtimes_7D_4 % in TeX
G:=Group("C51:7D4"); // GroupNames label
G:=SmallGroup(408,29);
// by ID
G=gap.SmallGroup(408,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,61,323,9604]);
// Polycyclic
G:=Group<a,b,c|a^51=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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