metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52.C4, Dic13.13C23, Q8.(C13⋊C4), (Q8×C13).C4, D13⋊C8⋊3C2, C13⋊2(C8○D4), C52.6(C2×C4), D26.2(C2×C4), C13⋊C8.2C22, C52.C4⋊4C2, C26.9(C22×C4), D52⋊C2.3C2, (C4×D13).13C22, C4.6(C2×C13⋊C4), C2.10(C22×C13⋊C4), SmallGroup(416,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52.C4
G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, cac-1=a5, cbc-1=a4b >
Subgroups: 404 in 62 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, D4, Q8, C13, C2×C8, M4(2), C4○D4, D13, C26, C8○D4, Dic13, C52, D26, C13⋊C8, C13⋊C8, C4×D13, D52, Q8×C13, D13⋊C8, C52.C4, D52⋊C2, D52.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C8○D4, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, D52.C4
►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 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(69 104)(70 103)(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)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)(113 142)(114 141)(115 140)(116 139)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(151 156)(152 155)(153 154)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(183 208)(184 207)(185 206)(186 205)(187 204)(188 203)(189 202)(190 201)(191 200)(192 199)(193 198)(194 197)(195 196)
(1 128 61 170 27 154 87 196)(2 149 86 175 28 123 60 201)(3 118 59 180 29 144 85 206)(4 139 84 185 30 113 58 159)(5 108 57 190 31 134 83 164)(6 129 82 195 32 155 56 169)(7 150 55 200 33 124 81 174)(8 119 80 205 34 145 54 179)(9 140 53 158 35 114 79 184)(10 109 78 163 36 135 104 189)(11 130 103 168 37 156 77 194)(12 151 76 173 38 125 102 199)(13 120 101 178 39 146 75 204)(14 141 74 183 40 115 100 157)(15 110 99 188 41 136 73 162)(16 131 72 193 42 105 98 167)(17 152 97 198 43 126 71 172)(18 121 70 203 44 147 96 177)(19 142 95 208 45 116 69 182)(20 111 68 161 46 137 94 187)(21 132 93 166 47 106 67 192)(22 153 66 171 48 127 92 197)(23 122 91 176 49 148 65 202)(24 143 64 181 50 117 90 207)(25 112 89 186 51 138 63 160)(26 133 62 191 52 107 88 165)
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)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(69,104)(70,103)(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)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(151,156)(152,155)(153,154)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196), (1,128,61,170,27,154,87,196)(2,149,86,175,28,123,60,201)(3,118,59,180,29,144,85,206)(4,139,84,185,30,113,58,159)(5,108,57,190,31,134,83,164)(6,129,82,195,32,155,56,169)(7,150,55,200,33,124,81,174)(8,119,80,205,34,145,54,179)(9,140,53,158,35,114,79,184)(10,109,78,163,36,135,104,189)(11,130,103,168,37,156,77,194)(12,151,76,173,38,125,102,199)(13,120,101,178,39,146,75,204)(14,141,74,183,40,115,100,157)(15,110,99,188,41,136,73,162)(16,131,72,193,42,105,98,167)(17,152,97,198,43,126,71,172)(18,121,70,203,44,147,96,177)(19,142,95,208,45,116,69,182)(20,111,68,161,46,137,94,187)(21,132,93,166,47,106,67,192)(22,153,66,171,48,127,92,197)(23,122,91,176,49,148,65,202)(24,143,64,181,50,117,90,207)(25,112,89,186,51,138,63,160)(26,133,62,191,52,107,88,165)>;
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)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(69,104)(70,103)(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)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(151,156)(152,155)(153,154)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196), (1,128,61,170,27,154,87,196)(2,149,86,175,28,123,60,201)(3,118,59,180,29,144,85,206)(4,139,84,185,30,113,58,159)(5,108,57,190,31,134,83,164)(6,129,82,195,32,155,56,169)(7,150,55,200,33,124,81,174)(8,119,80,205,34,145,54,179)(9,140,53,158,35,114,79,184)(10,109,78,163,36,135,104,189)(11,130,103,168,37,156,77,194)(12,151,76,173,38,125,102,199)(13,120,101,178,39,146,75,204)(14,141,74,183,40,115,100,157)(15,110,99,188,41,136,73,162)(16,131,72,193,42,105,98,167)(17,152,97,198,43,126,71,172)(18,121,70,203,44,147,96,177)(19,142,95,208,45,116,69,182)(20,111,68,161,46,137,94,187)(21,132,93,166,47,106,67,192)(22,153,66,171,48,127,92,197)(23,122,91,176,49,148,65,202)(24,143,64,181,50,117,90,207)(25,112,89,186,51,138,63,160)(26,133,62,191,52,107,88,165) );
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),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(69,104),(70,103),(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),(105,150),(106,149),(107,148),(108,147),(109,146),(110,145),(111,144),(112,143),(113,142),(114,141),(115,140),(116,139),(117,138),(118,137),(119,136),(120,135),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(151,156),(152,155),(153,154),(157,182),(158,181),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(183,208),(184,207),(185,206),(186,205),(187,204),(188,203),(189,202),(190,201),(191,200),(192,199),(193,198),(194,197),(195,196)], [(1,128,61,170,27,154,87,196),(2,149,86,175,28,123,60,201),(3,118,59,180,29,144,85,206),(4,139,84,185,30,113,58,159),(5,108,57,190,31,134,83,164),(6,129,82,195,32,155,56,169),(7,150,55,200,33,124,81,174),(8,119,80,205,34,145,54,179),(9,140,53,158,35,114,79,184),(10,109,78,163,36,135,104,189),(11,130,103,168,37,156,77,194),(12,151,76,173,38,125,102,199),(13,120,101,178,39,146,75,204),(14,141,74,183,40,115,100,157),(15,110,99,188,41,136,73,162),(16,131,72,193,42,105,98,167),(17,152,97,198,43,126,71,172),(18,121,70,203,44,147,96,177),(19,142,95,208,45,116,69,182),(20,111,68,161,46,137,94,187),(21,132,93,166,47,106,67,192),(22,153,66,171,48,127,92,197),(23,122,91,176,49,148,65,202),(24,143,64,181,50,117,90,207),(25,112,89,186,51,138,63,160),(26,133,62,191,52,107,88,165)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52I |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 26 | 26 | 26 | 2 | 2 | 2 | 13 | 13 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8○D4 | C13⋊C4 | C2×C13⋊C4 | D52.C4 |
| kernel | D52.C4 | D13⋊C8 | C52.C4 | D52⋊C2 | D52 | Q8×C13 | C13 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 3 | 9 | 3 |
Matrix representation of D52.C4 ►in GL6(𝔽313)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 312 | 1 |
| 0 | 0 | 71 | 31 | 281 | 242 |
| 0 | 0 | 281 | 61 | 40 | 2 |
| 0 | 0 | 71 | 30 | 311 | 212 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 312 | 0 |
| 0 | 0 | 1 | 0 | 312 | 0 |
| 0 | 0 | 0 | 0 | 312 | 0 |
| 0 | 0 | 282 | 31 | 70 | 1 |
| 308 | 0 | 0 | 0 | 0 | 0 |
| 0 | 308 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 205 | 108 | 225 |
| 0 | 0 | 132 | 290 | 150 | 163 |
| 0 | 0 | 114 | 119 | 308 | 67 |
| 0 | 0 | 269 | 290 | 13 | 253 |
G:=sub<GL(6,GF(313))| [0,312,0,0,0,0,1,0,0,0,0,0,0,0,0,71,281,71,0,0,0,31,61,30,0,0,312,281,40,311,0,0,1,242,2,212],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,282,0,0,1,0,0,31,0,0,312,312,312,70,0,0,0,0,0,1],[308,0,0,0,0,0,0,308,0,0,0,0,0,0,88,132,114,269,0,0,205,290,119,290,0,0,108,150,308,13,0,0,225,163,67,253] >;
D52.C4 in GAP, Magma, Sage, TeX
D_{52}.C_4 % in TeX
G:=Group("D52.C4"); // GroupNames label
G:=SmallGroup(416,207);
// by ID
G=gap.SmallGroup(416,207);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,188,86,69,9221,1751]);
// 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^5,c*b*c^-1=a^4*b>;
// generators/relations