metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C53⋊2D4, C22⋊D53, Dic53⋊C2, D106⋊2C2, C2.5D106, C106.5C22, (C2×C106)⋊2C2, SmallGroup(424,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C53⋊D4
G = < a,b,c | a53=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C53⋊D4
►On 212 points
Generators in S212
(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)(160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212)
(1 123 81 203)(2 122 82 202)(3 121 83 201)(4 120 84 200)(5 119 85 199)(6 118 86 198)(7 117 87 197)(8 116 88 196)(9 115 89 195)(10 114 90 194)(11 113 91 193)(12 112 92 192)(13 111 93 191)(14 110 94 190)(15 109 95 189)(16 108 96 188)(17 107 97 187)(18 159 98 186)(19 158 99 185)(20 157 100 184)(21 156 101 183)(22 155 102 182)(23 154 103 181)(24 153 104 180)(25 152 105 179)(26 151 106 178)(27 150 54 177)(28 149 55 176)(29 148 56 175)(30 147 57 174)(31 146 58 173)(32 145 59 172)(33 144 60 171)(34 143 61 170)(35 142 62 169)(36 141 63 168)(37 140 64 167)(38 139 65 166)(39 138 66 165)(40 137 67 164)(41 136 68 163)(42 135 69 162)(43 134 70 161)(44 133 71 160)(45 132 72 212)(46 131 73 211)(47 130 74 210)(48 129 75 209)(49 128 76 208)(50 127 77 207)(51 126 78 206)(52 125 79 205)(53 124 80 204)
(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(54 55)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(107 166)(108 165)(109 164)(110 163)(111 162)(112 161)(113 160)(114 212)(115 211)(116 210)(117 209)(118 208)(119 207)(120 206)(121 205)(122 204)(123 203)(124 202)(125 201)(126 200)(127 199)(128 198)(129 197)(130 196)(131 195)(132 194)(133 193)(134 192)(135 191)(136 190)(137 189)(138 188)(139 187)(140 186)(141 185)(142 184)(143 183)(144 182)(145 181)(146 180)(147 179)(148 178)(149 177)(150 176)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)
G:=sub<Sym(212)| (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)(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,123,81,203)(2,122,82,202)(3,121,83,201)(4,120,84,200)(5,119,85,199)(6,118,86,198)(7,117,87,197)(8,116,88,196)(9,115,89,195)(10,114,90,194)(11,113,91,193)(12,112,92,192)(13,111,93,191)(14,110,94,190)(15,109,95,189)(16,108,96,188)(17,107,97,187)(18,159,98,186)(19,158,99,185)(20,157,100,184)(21,156,101,183)(22,155,102,182)(23,154,103,181)(24,153,104,180)(25,152,105,179)(26,151,106,178)(27,150,54,177)(28,149,55,176)(29,148,56,175)(30,147,57,174)(31,146,58,173)(32,145,59,172)(33,144,60,171)(34,143,61,170)(35,142,62,169)(36,141,63,168)(37,140,64,167)(38,139,65,166)(39,138,66,165)(40,137,67,164)(41,136,68,163)(42,135,69,162)(43,134,70,161)(44,133,71,160)(45,132,72,212)(46,131,73,211)(47,130,74,210)(48,129,75,209)(49,128,76,208)(50,127,77,207)(51,126,78,206)(52,125,79,205)(53,124,80,204), (2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(54,55)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,212)(115,211)(116,210)(117,209)(118,208)(119,207)(120,206)(121,205)(122,204)(123,203)(124,202)(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)(132,194)(133,193)(134,192)(135,191)(136,190)(137,189)(138,188)(139,187)(140,186)(141,185)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)>;
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)(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,123,81,203)(2,122,82,202)(3,121,83,201)(4,120,84,200)(5,119,85,199)(6,118,86,198)(7,117,87,197)(8,116,88,196)(9,115,89,195)(10,114,90,194)(11,113,91,193)(12,112,92,192)(13,111,93,191)(14,110,94,190)(15,109,95,189)(16,108,96,188)(17,107,97,187)(18,159,98,186)(19,158,99,185)(20,157,100,184)(21,156,101,183)(22,155,102,182)(23,154,103,181)(24,153,104,180)(25,152,105,179)(26,151,106,178)(27,150,54,177)(28,149,55,176)(29,148,56,175)(30,147,57,174)(31,146,58,173)(32,145,59,172)(33,144,60,171)(34,143,61,170)(35,142,62,169)(36,141,63,168)(37,140,64,167)(38,139,65,166)(39,138,66,165)(40,137,67,164)(41,136,68,163)(42,135,69,162)(43,134,70,161)(44,133,71,160)(45,132,72,212)(46,131,73,211)(47,130,74,210)(48,129,75,209)(49,128,76,208)(50,127,77,207)(51,126,78,206)(52,125,79,205)(53,124,80,204), (2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(54,55)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,212)(115,211)(116,210)(117,209)(118,208)(119,207)(120,206)(121,205)(122,204)(123,203)(124,202)(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)(132,194)(133,193)(134,192)(135,191)(136,190)(137,189)(138,188)(139,187)(140,186)(141,185)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167) );
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),(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212)], [(1,123,81,203),(2,122,82,202),(3,121,83,201),(4,120,84,200),(5,119,85,199),(6,118,86,198),(7,117,87,197),(8,116,88,196),(9,115,89,195),(10,114,90,194),(11,113,91,193),(12,112,92,192),(13,111,93,191),(14,110,94,190),(15,109,95,189),(16,108,96,188),(17,107,97,187),(18,159,98,186),(19,158,99,185),(20,157,100,184),(21,156,101,183),(22,155,102,182),(23,154,103,181),(24,153,104,180),(25,152,105,179),(26,151,106,178),(27,150,54,177),(28,149,55,176),(29,148,56,175),(30,147,57,174),(31,146,58,173),(32,145,59,172),(33,144,60,171),(34,143,61,170),(35,142,62,169),(36,141,63,168),(37,140,64,167),(38,139,65,166),(39,138,66,165),(40,137,67,164),(41,136,68,163),(42,135,69,162),(43,134,70,161),(44,133,71,160),(45,132,72,212),(46,131,73,211),(47,130,74,210),(48,129,75,209),(49,128,76,208),(50,127,77,207),(51,126,78,206),(52,125,79,205),(53,124,80,204)], [(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(54,55),(56,106),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(107,166),(108,165),(109,164),(110,163),(111,162),(112,161),(113,160),(114,212),(115,211),(116,210),(117,209),(118,208),(119,207),(120,206),(121,205),(122,204),(123,203),(124,202),(125,201),(126,200),(127,199),(128,198),(129,197),(130,196),(131,195),(132,194),(133,193),(134,192),(135,191),(136,190),(137,189),(138,188),(139,187),(140,186),(141,185),(142,184),(143,183),(144,182),(145,181),(146,180),(147,179),(148,178),(149,177),(150,176),(151,175),(152,174),(153,173),(154,172),(155,171),(156,170),(157,169),(158,168),(159,167)]])
109 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 53A | ··· | 53Z | 106A | ··· | 106BZ |
| order | 1 | 2 | 2 | 2 | 4 | 53 | ··· | 53 | 106 | ··· | 106 |
| size | 1 | 1 | 2 | 106 | 106 | 2 | ··· | 2 | 2 | ··· | 2 |
109 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D53 | D106 | C53⋊D4 |
| kernel | C53⋊D4 | Dic53 | D106 | C2×C106 | C53 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 26 | 26 | 52 |
Matrix representation of C53⋊D4 ►in GL2(𝔽1061) generated by
| 0 | 1 |
| 1060 | 1012 |
| 198 | 138 |
| 1046 | 863 |
| 1 | 0 |
| 1012 | 1060 |
G:=sub<GL(2,GF(1061))| [0,1060,1,1012],[198,1046,138,863],[1,1012,0,1060] >;
C53⋊D4 in GAP, Magma, Sage, TeX
C_{53}\rtimes D_4 % in TeX
G:=Group("C53:D4"); // GroupNames label
G:=SmallGroup(424,8);
// by ID
G=gap.SmallGroup(424,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,49,6659]);
// Polycyclic
G:=Group<a,b,c|a^53=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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