metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52.2C4, C8.12D26, M4(2)⋊5D13, C52.39C23, Dic26.2C4, C104.12C22, (C8×D13)⋊8C2, C13⋊4(C8○D4), C4.5(C4×D13), C8⋊D13⋊6C2, C52.34(C2×C4), D26.4(C2×C4), (C2×C4).46D26, C13⋊D4.2C4, C22.1(C4×D13), D52⋊5C2.3C2, (C13×M4(2))⋊4C2, C26.29(C22×C4), (C2×C52).26C22, Dic13.6(C2×C4), C4.39(C22×D13), C13⋊2C8.12C22, (C4×D13).24C22, (C2×C13⋊2C8)⋊3C2, C2.17(C2×C4×D13), (C2×C26).26(C2×C4), SmallGroup(416,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52.2C4
G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, cac-1=a27, cbc-1=a26b >
Subgroups: 344 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C13, C2×C8, M4(2), M4(2), C4○D4, D13, C26, C26, C8○D4, Dic13, C52, D26, C2×C26, C13⋊2C8, C104, Dic26, C4×D13, D52, C13⋊D4, C2×C52, C8×D13, C8⋊D13, C2×C13⋊2C8, C13×M4(2), D52⋊5C2, D52.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D13, C8○D4, D26, C4×D13, C22×D13, C2×C4×D13, D52.2C4
►On 208 points
Generators in S208
(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 52)(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)(53 60)(54 59)(55 58)(56 57)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)(127 156)(128 155)(129 154)(130 153)(131 152)(132 151)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(157 206)(158 205)(159 204)(160 203)(161 202)(162 201)(163 200)(164 199)(165 198)(166 197)(167 196)(168 195)(169 194)(170 193)(171 192)(172 191)(173 190)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(207 208)
(1 57 142 182 27 83 116 208)(2 84 143 157 28 58 117 183)(3 59 144 184 29 85 118 158)(4 86 145 159 30 60 119 185)(5 61 146 186 31 87 120 160)(6 88 147 161 32 62 121 187)(7 63 148 188 33 89 122 162)(8 90 149 163 34 64 123 189)(9 65 150 190 35 91 124 164)(10 92 151 165 36 66 125 191)(11 67 152 192 37 93 126 166)(12 94 153 167 38 68 127 193)(13 69 154 194 39 95 128 168)(14 96 155 169 40 70 129 195)(15 71 156 196 41 97 130 170)(16 98 105 171 42 72 131 197)(17 73 106 198 43 99 132 172)(18 100 107 173 44 74 133 199)(19 75 108 200 45 101 134 174)(20 102 109 175 46 76 135 201)(21 77 110 202 47 103 136 176)(22 104 111 177 48 78 137 203)(23 79 112 204 49 53 138 178)(24 54 113 179 50 80 139 205)(25 81 114 206 51 55 140 180)(26 56 115 181 52 82 141 207)
G:=sub<Sym(208)| (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), (1,52)(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)(53,60)(54,59)(55,58)(56,57)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(157,206)(158,205)(159,204)(160,203)(161,202)(162,201)(163,200)(164,199)(165,198)(166,197)(167,196)(168,195)(169,194)(170,193)(171,192)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)(207,208), (1,57,142,182,27,83,116,208)(2,84,143,157,28,58,117,183)(3,59,144,184,29,85,118,158)(4,86,145,159,30,60,119,185)(5,61,146,186,31,87,120,160)(6,88,147,161,32,62,121,187)(7,63,148,188,33,89,122,162)(8,90,149,163,34,64,123,189)(9,65,150,190,35,91,124,164)(10,92,151,165,36,66,125,191)(11,67,152,192,37,93,126,166)(12,94,153,167,38,68,127,193)(13,69,154,194,39,95,128,168)(14,96,155,169,40,70,129,195)(15,71,156,196,41,97,130,170)(16,98,105,171,42,72,131,197)(17,73,106,198,43,99,132,172)(18,100,107,173,44,74,133,199)(19,75,108,200,45,101,134,174)(20,102,109,175,46,76,135,201)(21,77,110,202,47,103,136,176)(22,104,111,177,48,78,137,203)(23,79,112,204,49,53,138,178)(24,54,113,179,50,80,139,205)(25,81,114,206,51,55,140,180)(26,56,115,181,52,82,141,207)>;
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), (1,52)(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)(53,60)(54,59)(55,58)(56,57)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(157,206)(158,205)(159,204)(160,203)(161,202)(162,201)(163,200)(164,199)(165,198)(166,197)(167,196)(168,195)(169,194)(170,193)(171,192)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)(207,208), (1,57,142,182,27,83,116,208)(2,84,143,157,28,58,117,183)(3,59,144,184,29,85,118,158)(4,86,145,159,30,60,119,185)(5,61,146,186,31,87,120,160)(6,88,147,161,32,62,121,187)(7,63,148,188,33,89,122,162)(8,90,149,163,34,64,123,189)(9,65,150,190,35,91,124,164)(10,92,151,165,36,66,125,191)(11,67,152,192,37,93,126,166)(12,94,153,167,38,68,127,193)(13,69,154,194,39,95,128,168)(14,96,155,169,40,70,129,195)(15,71,156,196,41,97,130,170)(16,98,105,171,42,72,131,197)(17,73,106,198,43,99,132,172)(18,100,107,173,44,74,133,199)(19,75,108,200,45,101,134,174)(20,102,109,175,46,76,135,201)(21,77,110,202,47,103,136,176)(22,104,111,177,48,78,137,203)(23,79,112,204,49,53,138,178)(24,54,113,179,50,80,139,205)(25,81,114,206,51,55,140,180)(26,56,115,181,52,82,141,207) );
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)], [(1,52),(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),(53,60),(54,59),(55,58),(56,57),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(111,120),(112,119),(113,118),(114,117),(115,116),(127,156),(128,155),(129,154),(130,153),(131,152),(132,151),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143),(141,142),(157,206),(158,205),(159,204),(160,203),(161,202),(162,201),(163,200),(164,199),(165,198),(166,197),(167,196),(168,195),(169,194),(170,193),(171,192),(172,191),(173,190),(174,189),(175,188),(176,187),(177,186),(178,185),(179,184),(180,183),(181,182),(207,208)], [(1,57,142,182,27,83,116,208),(2,84,143,157,28,58,117,183),(3,59,144,184,29,85,118,158),(4,86,145,159,30,60,119,185),(5,61,146,186,31,87,120,160),(6,88,147,161,32,62,121,187),(7,63,148,188,33,89,122,162),(8,90,149,163,34,64,123,189),(9,65,150,190,35,91,124,164),(10,92,151,165,36,66,125,191),(11,67,152,192,37,93,126,166),(12,94,153,167,38,68,127,193),(13,69,154,194,39,95,128,168),(14,96,155,169,40,70,129,195),(15,71,156,196,41,97,130,170),(16,98,105,171,42,72,131,197),(17,73,106,198,43,99,132,172),(18,100,107,173,44,74,133,199),(19,75,108,200,45,101,134,174),(20,102,109,175,46,76,135,201),(21,77,110,202,47,103,136,176),(22,104,111,177,48,78,137,203),(23,79,112,204,49,53,138,178),(24,54,113,179,50,80,139,205),(25,81,114,206,51,55,140,180),(26,56,115,181,52,82,141,207)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 26 | 26 | 1 | 1 | 2 | 26 | 26 | 2 | 2 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D13 | C8○D4 | D26 | D26 | C4×D13 | C4×D13 | D52.2C4 |
| kernel | D52.2C4 | C8×D13 | C8⋊D13 | C2×C13⋊2C8 | C13×M4(2) | D52⋊5C2 | Dic26 | D52 | C13⋊D4 | M4(2) | C13 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 4 | 12 | 6 | 12 | 12 | 12 |
Matrix representation of D52.2C4 ►in GL4(𝔽313) generated by
| 293 | 78 | 0 | 0 |
| 81 | 44 | 0 | 0 |
| 0 | 0 | 131 | 196 |
| 0 | 0 | 37 | 182 |
| 312 | 0 | 0 | 0 |
| 256 | 1 | 0 | 0 |
| 0 | 0 | 131 | 196 |
| 0 | 0 | 251 | 182 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 125 | 0 |
| 0 | 0 | 98 | 188 |
G:=sub<GL(4,GF(313))| [293,81,0,0,78,44,0,0,0,0,131,37,0,0,196,182],[312,256,0,0,0,1,0,0,0,0,131,251,0,0,196,182],[312,0,0,0,0,312,0,0,0,0,125,98,0,0,0,188] >;
D52.2C4 in GAP, Magma, Sage, TeX
D_{52}._2C_4 % in TeX
G:=Group("D52.2C4"); // GroupNames label
G:=SmallGroup(416,128);
// by ID
G=gap.SmallGroup(416,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,188,50,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=1,c^4=a^26,b*a*b=a^-1,c*a*c^-1=a^27,c*b*c^-1=a^26*b>;
// generators/relations