direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C53, C4⋊C106, C212⋊3C2, C22⋊C106, C106.6C22, (C2×C106)⋊1C2, C2.1(C2×C106), SmallGroup(424,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C53
G = < a,b,c | a53=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C53
►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)
(1 206 151 74)(2 207 152 75)(3 208 153 76)(4 209 154 77)(5 210 155 78)(6 211 156 79)(7 212 157 80)(8 160 158 81)(9 161 159 82)(10 162 107 83)(11 163 108 84)(12 164 109 85)(13 165 110 86)(14 166 111 87)(15 167 112 88)(16 168 113 89)(17 169 114 90)(18 170 115 91)(19 171 116 92)(20 172 117 93)(21 173 118 94)(22 174 119 95)(23 175 120 96)(24 176 121 97)(25 177 122 98)(26 178 123 99)(27 179 124 100)(28 180 125 101)(29 181 126 102)(30 182 127 103)(31 183 128 104)(32 184 129 105)(33 185 130 106)(34 186 131 54)(35 187 132 55)(36 188 133 56)(37 189 134 57)(38 190 135 58)(39 191 136 59)(40 192 137 60)(41 193 138 61)(42 194 139 62)(43 195 140 63)(44 196 141 64)(45 197 142 65)(46 198 143 66)(47 199 144 67)(48 200 145 68)(49 201 146 69)(50 202 147 70)(51 203 148 71)(52 204 149 72)(53 205 150 73)
(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 197)(66 198)(67 199)(68 200)(69 201)(70 202)(71 203)(72 204)(73 205)(74 206)(75 207)(76 208)(77 209)(78 210)(79 211)(80 212)(81 160)(82 161)(83 162)(84 163)(85 164)(86 165)(87 166)(88 167)(89 168)(90 169)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 178)(100 179)(101 180)(102 181)(103 182)(104 183)(105 184)(106 185)
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), (1,206,151,74)(2,207,152,75)(3,208,153,76)(4,209,154,77)(5,210,155,78)(6,211,156,79)(7,212,157,80)(8,160,158,81)(9,161,159,82)(10,162,107,83)(11,163,108,84)(12,164,109,85)(13,165,110,86)(14,166,111,87)(15,167,112,88)(16,168,113,89)(17,169,114,90)(18,170,115,91)(19,171,116,92)(20,172,117,93)(21,173,118,94)(22,174,119,95)(23,175,120,96)(24,176,121,97)(25,177,122,98)(26,178,123,99)(27,179,124,100)(28,180,125,101)(29,181,126,102)(30,182,127,103)(31,183,128,104)(32,184,129,105)(33,185,130,106)(34,186,131,54)(35,187,132,55)(36,188,133,56)(37,189,134,57)(38,190,135,58)(39,191,136,59)(40,192,137,60)(41,193,138,61)(42,194,139,62)(43,195,140,63)(44,196,141,64)(45,197,142,65)(46,198,143,66)(47,199,144,67)(48,200,145,68)(49,201,146,69)(50,202,147,70)(51,203,148,71)(52,204,149,72)(53,205,150,73), (54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,160)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185)>;
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), (1,206,151,74)(2,207,152,75)(3,208,153,76)(4,209,154,77)(5,210,155,78)(6,211,156,79)(7,212,157,80)(8,160,158,81)(9,161,159,82)(10,162,107,83)(11,163,108,84)(12,164,109,85)(13,165,110,86)(14,166,111,87)(15,167,112,88)(16,168,113,89)(17,169,114,90)(18,170,115,91)(19,171,116,92)(20,172,117,93)(21,173,118,94)(22,174,119,95)(23,175,120,96)(24,176,121,97)(25,177,122,98)(26,178,123,99)(27,179,124,100)(28,180,125,101)(29,181,126,102)(30,182,127,103)(31,183,128,104)(32,184,129,105)(33,185,130,106)(34,186,131,54)(35,187,132,55)(36,188,133,56)(37,189,134,57)(38,190,135,58)(39,191,136,59)(40,192,137,60)(41,193,138,61)(42,194,139,62)(43,195,140,63)(44,196,141,64)(45,197,142,65)(46,198,143,66)(47,199,144,67)(48,200,145,68)(49,201,146,69)(50,202,147,70)(51,203,148,71)(52,204,149,72)(53,205,150,73), (54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,160)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185) );
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)], [(1,206,151,74),(2,207,152,75),(3,208,153,76),(4,209,154,77),(5,210,155,78),(6,211,156,79),(7,212,157,80),(8,160,158,81),(9,161,159,82),(10,162,107,83),(11,163,108,84),(12,164,109,85),(13,165,110,86),(14,166,111,87),(15,167,112,88),(16,168,113,89),(17,169,114,90),(18,170,115,91),(19,171,116,92),(20,172,117,93),(21,173,118,94),(22,174,119,95),(23,175,120,96),(24,176,121,97),(25,177,122,98),(26,178,123,99),(27,179,124,100),(28,180,125,101),(29,181,126,102),(30,182,127,103),(31,183,128,104),(32,184,129,105),(33,185,130,106),(34,186,131,54),(35,187,132,55),(36,188,133,56),(37,189,134,57),(38,190,135,58),(39,191,136,59),(40,192,137,60),(41,193,138,61),(42,194,139,62),(43,195,140,63),(44,196,141,64),(45,197,142,65),(46,198,143,66),(47,199,144,67),(48,200,145,68),(49,201,146,69),(50,202,147,70),(51,203,148,71),(52,204,149,72),(53,205,150,73)], [(54,186),(55,187),(56,188),(57,189),(58,190),(59,191),(60,192),(61,193),(62,194),(63,195),(64,196),(65,197),(66,198),(67,199),(68,200),(69,201),(70,202),(71,203),(72,204),(73,205),(74,206),(75,207),(76,208),(77,209),(78,210),(79,211),(80,212),(81,160),(82,161),(83,162),(84,163),(85,164),(86,165),(87,166),(88,167),(89,168),(90,169),(91,170),(92,171),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,178),(100,179),(101,180),(102,181),(103,182),(104,183),(105,184),(106,185)]])
265 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 53A | ··· | 53AZ | 106A | ··· | 106AZ | 106BA | ··· | 106EZ | 212A | ··· | 212AZ |
| order | 1 | 2 | 2 | 2 | 4 | 53 | ··· | 53 | 106 | ··· | 106 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
265 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C53 | C106 | C106 | D4 | D4×C53 |
| kernel | D4×C53 | C212 | C2×C106 | D4 | C4 | C22 | C53 | C1 |
| # reps | 1 | 1 | 2 | 52 | 52 | 104 | 1 | 52 |
Matrix representation of D4×C53 ►in GL2(𝔽1061) generated by
| 978 | 0 |
| 0 | 978 |
| 1 | 2 |
| 1060 | 1060 |
| 1 | 0 |
| 1060 | 1060 |
G:=sub<GL(2,GF(1061))| [978,0,0,978],[1,1060,2,1060],[1,1060,0,1060] >;
D4×C53 in GAP, Magma, Sage, TeX
D_4\times C_{53} % in TeX
G:=Group("D4xC53"); // GroupNames label
G:=SmallGroup(424,10);
// by ID
G=gap.SmallGroup(424,10);
# by ID
G:=PCGroup([4,-2,-2,-53,-2,1713]);
// Polycyclic
G:=Group<a,b,c|a^53=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export