metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D52, C23.16D26, C52⋊3C4⋊5C2, C2.8(C2×D52), (C2×C26).4D4, C26.6(C2×D4), (C2×C4).9D26, C22⋊C4⋊6D13, D26⋊C4⋊7C2, (C2×C52).3C22, C26.23(C4○D4), (C2×C26).27C23, (C22×Dic13)⋊2C2, C13⋊2(C22.D4), C2.10(D4⋊2D13), (C22×C26).16C22, (C22×D13).5C22, C22.45(C22×D13), (C2×Dic13).31C22, (C13×C22⋊C4)⋊4C2, (C2×C13⋊D4).5C2, SmallGroup(416,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D52
G = < a,b,c,d | a2=b2=c52=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 536 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C26, C26, C22.D4, Dic13, C52, D26, C2×C26, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×C26, C52⋊3C4, D26⋊C4, C13×C22⋊C4, C22×Dic13, C2×C13⋊D4, C22.D52
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C22.D4, D26, D52, C22×D13, C2×D52, D4⋊2D13, C22.D52
►On 208 points
Generators in S208
(2 152)(4 154)(6 156)(8 106)(10 108)(12 110)(14 112)(16 114)(18 116)(20 118)(22 120)(24 122)(26 124)(28 126)(30 128)(32 130)(34 132)(36 134)(38 136)(40 138)(42 140)(44 142)(46 144)(48 146)(50 148)(52 150)(53 175)(55 177)(57 179)(59 181)(61 183)(63 185)(65 187)(67 189)(69 191)(71 193)(73 195)(75 197)(77 199)(79 201)(81 203)(83 205)(85 207)(87 157)(89 159)(91 161)(93 163)(95 165)(97 167)(99 169)(101 171)(103 173)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 175)(54 176)(55 177)(56 178)(57 179)(58 180)(59 181)(60 182)(61 183)(62 184)(63 185)(64 186)(65 187)(66 188)(67 189)(68 190)(69 191)(70 192)(71 193)(72 194)(73 195)(74 196)(75 197)(76 198)(77 199)(78 200)(79 201)(80 202)(81 203)(82 204)(83 205)(84 206)(85 207)(86 208)(87 157)(88 158)(89 159)(90 160)(91 161)(92 162)(93 163)(94 164)(95 165)(96 166)(97 167)(98 168)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)
(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)
(1 180 151 58)(2 57 152 179)(3 178 153 56)(4 55 154 177)(5 176 155 54)(6 53 156 175)(7 174 105 104)(8 103 106 173)(9 172 107 102)(10 101 108 171)(11 170 109 100)(12 99 110 169)(13 168 111 98)(14 97 112 167)(15 166 113 96)(16 95 114 165)(17 164 115 94)(18 93 116 163)(19 162 117 92)(20 91 118 161)(21 160 119 90)(22 89 120 159)(23 158 121 88)(24 87 122 157)(25 208 123 86)(26 85 124 207)(27 206 125 84)(28 83 126 205)(29 204 127 82)(30 81 128 203)(31 202 129 80)(32 79 130 201)(33 200 131 78)(34 77 132 199)(35 198 133 76)(36 75 134 197)(37 196 135 74)(38 73 136 195)(39 194 137 72)(40 71 138 193)(41 192 139 70)(42 69 140 191)(43 190 141 68)(44 67 142 189)(45 188 143 66)(46 65 144 187)(47 186 145 64)(48 63 146 185)(49 184 147 62)(50 61 148 183)(51 182 149 60)(52 59 150 181)
G:=sub<Sym(208)| (2,152)(4,154)(6,156)(8,106)(10,108)(12,110)(14,112)(16,114)(18,116)(20,118)(22,120)(24,122)(26,124)(28,126)(30,128)(32,130)(34,132)(36,134)(38,136)(40,138)(42,140)(44,142)(46,144)(48,146)(50,148)(52,150)(53,175)(55,177)(57,179)(59,181)(61,183)(63,185)(65,187)(67,189)(69,191)(71,193)(73,195)(75,197)(77,199)(79,201)(81,203)(83,205)(85,207)(87,157)(89,159)(91,161)(93,163)(95,165)(97,167)(99,169)(101,171)(103,173), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,181)(60,182)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,197)(76,198)(77,199)(78,200)(79,201)(80,202)(81,203)(82,204)(83,205)(84,206)(85,207)(86,208)(87,157)(88,158)(89,159)(90,160)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174), (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), (1,180,151,58)(2,57,152,179)(3,178,153,56)(4,55,154,177)(5,176,155,54)(6,53,156,175)(7,174,105,104)(8,103,106,173)(9,172,107,102)(10,101,108,171)(11,170,109,100)(12,99,110,169)(13,168,111,98)(14,97,112,167)(15,166,113,96)(16,95,114,165)(17,164,115,94)(18,93,116,163)(19,162,117,92)(20,91,118,161)(21,160,119,90)(22,89,120,159)(23,158,121,88)(24,87,122,157)(25,208,123,86)(26,85,124,207)(27,206,125,84)(28,83,126,205)(29,204,127,82)(30,81,128,203)(31,202,129,80)(32,79,130,201)(33,200,131,78)(34,77,132,199)(35,198,133,76)(36,75,134,197)(37,196,135,74)(38,73,136,195)(39,194,137,72)(40,71,138,193)(41,192,139,70)(42,69,140,191)(43,190,141,68)(44,67,142,189)(45,188,143,66)(46,65,144,187)(47,186,145,64)(48,63,146,185)(49,184,147,62)(50,61,148,183)(51,182,149,60)(52,59,150,181)>;
G:=Group( (2,152)(4,154)(6,156)(8,106)(10,108)(12,110)(14,112)(16,114)(18,116)(20,118)(22,120)(24,122)(26,124)(28,126)(30,128)(32,130)(34,132)(36,134)(38,136)(40,138)(42,140)(44,142)(46,144)(48,146)(50,148)(52,150)(53,175)(55,177)(57,179)(59,181)(61,183)(63,185)(65,187)(67,189)(69,191)(71,193)(73,195)(75,197)(77,199)(79,201)(81,203)(83,205)(85,207)(87,157)(89,159)(91,161)(93,163)(95,165)(97,167)(99,169)(101,171)(103,173), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,181)(60,182)(61,183)(62,184)(63,185)(64,186)(65,187)(66,188)(67,189)(68,190)(69,191)(70,192)(71,193)(72,194)(73,195)(74,196)(75,197)(76,198)(77,199)(78,200)(79,201)(80,202)(81,203)(82,204)(83,205)(84,206)(85,207)(86,208)(87,157)(88,158)(89,159)(90,160)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174), (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), (1,180,151,58)(2,57,152,179)(3,178,153,56)(4,55,154,177)(5,176,155,54)(6,53,156,175)(7,174,105,104)(8,103,106,173)(9,172,107,102)(10,101,108,171)(11,170,109,100)(12,99,110,169)(13,168,111,98)(14,97,112,167)(15,166,113,96)(16,95,114,165)(17,164,115,94)(18,93,116,163)(19,162,117,92)(20,91,118,161)(21,160,119,90)(22,89,120,159)(23,158,121,88)(24,87,122,157)(25,208,123,86)(26,85,124,207)(27,206,125,84)(28,83,126,205)(29,204,127,82)(30,81,128,203)(31,202,129,80)(32,79,130,201)(33,200,131,78)(34,77,132,199)(35,198,133,76)(36,75,134,197)(37,196,135,74)(38,73,136,195)(39,194,137,72)(40,71,138,193)(41,192,139,70)(42,69,140,191)(43,190,141,68)(44,67,142,189)(45,188,143,66)(46,65,144,187)(47,186,145,64)(48,63,146,185)(49,184,147,62)(50,61,148,183)(51,182,149,60)(52,59,150,181) );
G=PermutationGroup([[(2,152),(4,154),(6,156),(8,106),(10,108),(12,110),(14,112),(16,114),(18,116),(20,118),(22,120),(24,122),(26,124),(28,126),(30,128),(32,130),(34,132),(36,134),(38,136),(40,138),(42,140),(44,142),(46,144),(48,146),(50,148),(52,150),(53,175),(55,177),(57,179),(59,181),(61,183),(63,185),(65,187),(67,189),(69,191),(71,193),(73,195),(75,197),(77,199),(79,201),(81,203),(83,205),(85,207),(87,157),(89,159),(91,161),(93,163),(95,165),(97,167),(99,169),(101,171),(103,173)], [(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,105),(8,106),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,113),(16,114),(17,115),(18,116),(19,117),(20,118),(21,119),(22,120),(23,121),(24,122),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,129),(32,130),(33,131),(34,132),(35,133),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,141),(44,142),(45,143),(46,144),(47,145),(48,146),(49,147),(50,148),(51,149),(52,150),(53,175),(54,176),(55,177),(56,178),(57,179),(58,180),(59,181),(60,182),(61,183),(62,184),(63,185),(64,186),(65,187),(66,188),(67,189),(68,190),(69,191),(70,192),(71,193),(72,194),(73,195),(74,196),(75,197),(76,198),(77,199),(78,200),(79,201),(80,202),(81,203),(82,204),(83,205),(84,206),(85,207),(86,208),(87,157),(88,158),(89,159),(90,160),(91,161),(92,162),(93,163),(94,164),(95,165),(96,166),(97,167),(98,168),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174)], [(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)], [(1,180,151,58),(2,57,152,179),(3,178,153,56),(4,55,154,177),(5,176,155,54),(6,53,156,175),(7,174,105,104),(8,103,106,173),(9,172,107,102),(10,101,108,171),(11,170,109,100),(12,99,110,169),(13,168,111,98),(14,97,112,167),(15,166,113,96),(16,95,114,165),(17,164,115,94),(18,93,116,163),(19,162,117,92),(20,91,118,161),(21,160,119,90),(22,89,120,159),(23,158,121,88),(24,87,122,157),(25,208,123,86),(26,85,124,207),(27,206,125,84),(28,83,126,205),(29,204,127,82),(30,81,128,203),(31,202,129,80),(32,79,130,201),(33,200,131,78),(34,77,132,199),(35,198,133,76),(36,75,134,197),(37,196,135,74),(38,73,136,195),(39,194,137,72),(40,71,138,193),(41,192,139,70),(42,69,140,191),(43,190,141,68),(44,67,142,189),(45,188,143,66),(46,65,144,187),(47,186,145,64),(48,63,146,185),(49,184,147,62),(50,61,148,183),(51,182,149,60),(52,59,150,181)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 52 | 4 | 4 | 26 | 26 | 26 | 26 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | D26 | D52 | D4⋊2D13 |
| kernel | C22.D52 | C52⋊3C4 | D26⋊C4 | C13×C22⋊C4 | C22×Dic13 | C2×C13⋊D4 | C2×C26 | C26 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 12 | 6 | 24 | 12 |
Matrix representation of C22.D52 ►in GL4(𝔽53) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 30 | 2 | 0 | 0 |
| 44 | 33 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 52 | 0 |
| 24 | 14 | 0 | 0 |
| 46 | 29 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[30,44,0,0,2,33,0,0,0,0,0,52,0,0,1,0],[24,46,0,0,14,29,0,0,0,0,23,0,0,0,0,23] >;
C22.D52 in GAP, Magma, Sage, TeX
C_2^2.D_{52} % in TeX
G:=Group("C2^2.D52"); // GroupNames label
G:=SmallGroup(416,107);
// by ID
G=gap.SmallGroup(416,107);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,218,188,122,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^52=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations