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G = D262Q8order 416 = 25·13

2nd semidirect product of D26 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D262Q8, C4.13D52, C52.11D4, C4⋊C45D13, C523C46C2, C26.8(C2×D4), C2.7(Q8×D13), C2.10(C2×D52), (C2×C4).14D26, C133(C22⋊Q8), C26.14(C2×Q8), (C2×Dic26)⋊7C2, (C2×C52).6C22, D26⋊C4.3C2, C26.27(C4○D4), (C2×C26).38C23, C2.13(D42D13), C22.52(C22×D13), (C2×Dic13).13C22, (C22×D13).25C22, (C13×C4⋊C4)⋊8C2, (C2×C4×D13).3C2, SmallGroup(416,118)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D262Q8
C1C13C26C2×C26C22×D13C2×C4×D13 — D262Q8
C13C2×C26 — D262Q8
C1C22C4⋊C4

Generators and relations for D262Q8
 G = < a,b,c,d | a26=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a11b, bd=db, dcd-1=c-1 >

Subgroups: 520 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, D13 [×2], C26 [×3], C22⋊Q8, Dic13 [×3], C52 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, Dic26 [×2], C4×D13 [×2], C2×Dic13, C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13, C523C4 [×2], D26⋊C4 [×2], C13×C4⋊C4, C2×Dic26, C2×C4×D13, D262Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26 [×3], D52 [×2], C22×D13, C2×D52, D42D13, Q8×D13, D262Q8

Smallest permutation representation of D262Q8
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)
(1 164)(2 163)(3 162)(4 161)(5 160)(6 159)(7 158)(8 157)(9 182)(10 181)(11 180)(12 179)(13 178)(14 177)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 168)(24 167)(25 166)(26 165)(27 83)(28 82)(29 81)(30 80)(31 79)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 156)(62 155)(63 154)(64 153)(65 152)(66 151)(67 150)(68 149)(69 148)(70 147)(71 146)(72 145)(73 144)(74 143)(75 142)(76 141)(77 140)(78 139)(105 199)(106 198)(107 197)(108 196)(109 195)(110 194)(111 193)(112 192)(113 191)(114 190)(115 189)(116 188)(117 187)(118 186)(119 185)(120 184)(121 183)(122 208)(123 207)(124 206)(125 205)(126 204)(127 203)(128 202)(129 201)(130 200)
(1 39 165 85)(2 38 166 84)(3 37 167 83)(4 36 168 82)(5 35 169 81)(6 34 170 80)(7 33 171 79)(8 32 172 104)(9 31 173 103)(10 30 174 102)(11 29 175 101)(12 28 176 100)(13 27 177 99)(14 52 178 98)(15 51 179 97)(16 50 180 96)(17 49 181 95)(18 48 182 94)(19 47 157 93)(20 46 158 92)(21 45 159 91)(22 44 160 90)(23 43 161 89)(24 42 162 88)(25 41 163 87)(26 40 164 86)(53 109 150 198)(54 108 151 197)(55 107 152 196)(56 106 153 195)(57 105 154 194)(58 130 155 193)(59 129 156 192)(60 128 131 191)(61 127 132 190)(62 126 133 189)(63 125 134 188)(64 124 135 187)(65 123 136 186)(66 122 137 185)(67 121 138 184)(68 120 139 183)(69 119 140 208)(70 118 141 207)(71 117 142 206)(72 116 143 205)(73 115 144 204)(74 114 145 203)(75 113 146 202)(76 112 147 201)(77 111 148 200)(78 110 149 199)
(1 121 165 184)(2 122 166 185)(3 123 167 186)(4 124 168 187)(5 125 169 188)(6 126 170 189)(7 127 171 190)(8 128 172 191)(9 129 173 192)(10 130 174 193)(11 105 175 194)(12 106 176 195)(13 107 177 196)(14 108 178 197)(15 109 179 198)(16 110 180 199)(17 111 181 200)(18 112 182 201)(19 113 157 202)(20 114 158 203)(21 115 159 204)(22 116 160 205)(23 117 161 206)(24 118 162 207)(25 119 163 208)(26 120 164 183)(27 55 99 152)(28 56 100 153)(29 57 101 154)(30 58 102 155)(31 59 103 156)(32 60 104 131)(33 61 79 132)(34 62 80 133)(35 63 81 134)(36 64 82 135)(37 65 83 136)(38 66 84 137)(39 67 85 138)(40 68 86 139)(41 69 87 140)(42 70 88 141)(43 71 89 142)(44 72 90 143)(45 73 91 144)(46 74 92 145)(47 75 93 146)(48 76 94 147)(49 77 95 148)(50 78 96 149)(51 53 97 150)(52 54 98 151)

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), (1,164)(2,163)(3,162)(4,161)(5,160)(6,159)(7,158)(8,157)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,83)(28,82)(29,81)(30,80)(31,79)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(105,199)(106,198)(107,197)(108,196)(109,195)(110,194)(111,193)(112,192)(113,191)(114,190)(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)(122,208)(123,207)(124,206)(125,205)(126,204)(127,203)(128,202)(129,201)(130,200), (1,39,165,85)(2,38,166,84)(3,37,167,83)(4,36,168,82)(5,35,169,81)(6,34,170,80)(7,33,171,79)(8,32,172,104)(9,31,173,103)(10,30,174,102)(11,29,175,101)(12,28,176,100)(13,27,177,99)(14,52,178,98)(15,51,179,97)(16,50,180,96)(17,49,181,95)(18,48,182,94)(19,47,157,93)(20,46,158,92)(21,45,159,91)(22,44,160,90)(23,43,161,89)(24,42,162,88)(25,41,163,87)(26,40,164,86)(53,109,150,198)(54,108,151,197)(55,107,152,196)(56,106,153,195)(57,105,154,194)(58,130,155,193)(59,129,156,192)(60,128,131,191)(61,127,132,190)(62,126,133,189)(63,125,134,188)(64,124,135,187)(65,123,136,186)(66,122,137,185)(67,121,138,184)(68,120,139,183)(69,119,140,208)(70,118,141,207)(71,117,142,206)(72,116,143,205)(73,115,144,204)(74,114,145,203)(75,113,146,202)(76,112,147,201)(77,111,148,200)(78,110,149,199), (1,121,165,184)(2,122,166,185)(3,123,167,186)(4,124,168,187)(5,125,169,188)(6,126,170,189)(7,127,171,190)(8,128,172,191)(9,129,173,192)(10,130,174,193)(11,105,175,194)(12,106,176,195)(13,107,177,196)(14,108,178,197)(15,109,179,198)(16,110,180,199)(17,111,181,200)(18,112,182,201)(19,113,157,202)(20,114,158,203)(21,115,159,204)(22,116,160,205)(23,117,161,206)(24,118,162,207)(25,119,163,208)(26,120,164,183)(27,55,99,152)(28,56,100,153)(29,57,101,154)(30,58,102,155)(31,59,103,156)(32,60,104,131)(33,61,79,132)(34,62,80,133)(35,63,81,134)(36,64,82,135)(37,65,83,136)(38,66,84,137)(39,67,85,138)(40,68,86,139)(41,69,87,140)(42,70,88,141)(43,71,89,142)(44,72,90,143)(45,73,91,144)(46,74,92,145)(47,75,93,146)(48,76,94,147)(49,77,95,148)(50,78,96,149)(51,53,97,150)(52,54,98,151)>;

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), (1,164)(2,163)(3,162)(4,161)(5,160)(6,159)(7,158)(8,157)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,83)(28,82)(29,81)(30,80)(31,79)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(105,199)(106,198)(107,197)(108,196)(109,195)(110,194)(111,193)(112,192)(113,191)(114,190)(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)(122,208)(123,207)(124,206)(125,205)(126,204)(127,203)(128,202)(129,201)(130,200), (1,39,165,85)(2,38,166,84)(3,37,167,83)(4,36,168,82)(5,35,169,81)(6,34,170,80)(7,33,171,79)(8,32,172,104)(9,31,173,103)(10,30,174,102)(11,29,175,101)(12,28,176,100)(13,27,177,99)(14,52,178,98)(15,51,179,97)(16,50,180,96)(17,49,181,95)(18,48,182,94)(19,47,157,93)(20,46,158,92)(21,45,159,91)(22,44,160,90)(23,43,161,89)(24,42,162,88)(25,41,163,87)(26,40,164,86)(53,109,150,198)(54,108,151,197)(55,107,152,196)(56,106,153,195)(57,105,154,194)(58,130,155,193)(59,129,156,192)(60,128,131,191)(61,127,132,190)(62,126,133,189)(63,125,134,188)(64,124,135,187)(65,123,136,186)(66,122,137,185)(67,121,138,184)(68,120,139,183)(69,119,140,208)(70,118,141,207)(71,117,142,206)(72,116,143,205)(73,115,144,204)(74,114,145,203)(75,113,146,202)(76,112,147,201)(77,111,148,200)(78,110,149,199), (1,121,165,184)(2,122,166,185)(3,123,167,186)(4,124,168,187)(5,125,169,188)(6,126,170,189)(7,127,171,190)(8,128,172,191)(9,129,173,192)(10,130,174,193)(11,105,175,194)(12,106,176,195)(13,107,177,196)(14,108,178,197)(15,109,179,198)(16,110,180,199)(17,111,181,200)(18,112,182,201)(19,113,157,202)(20,114,158,203)(21,115,159,204)(22,116,160,205)(23,117,161,206)(24,118,162,207)(25,119,163,208)(26,120,164,183)(27,55,99,152)(28,56,100,153)(29,57,101,154)(30,58,102,155)(31,59,103,156)(32,60,104,131)(33,61,79,132)(34,62,80,133)(35,63,81,134)(36,64,82,135)(37,65,83,136)(38,66,84,137)(39,67,85,138)(40,68,86,139)(41,69,87,140)(42,70,88,141)(43,71,89,142)(44,72,90,143)(45,73,91,144)(46,74,92,145)(47,75,93,146)(48,76,94,147)(49,77,95,148)(50,78,96,149)(51,53,97,150)(52,54,98,151) );

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)], [(1,164),(2,163),(3,162),(4,161),(5,160),(6,159),(7,158),(8,157),(9,182),(10,181),(11,180),(12,179),(13,178),(14,177),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,168),(24,167),(25,166),(26,165),(27,83),(28,82),(29,81),(30,80),(31,79),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,156),(62,155),(63,154),(64,153),(65,152),(66,151),(67,150),(68,149),(69,148),(70,147),(71,146),(72,145),(73,144),(74,143),(75,142),(76,141),(77,140),(78,139),(105,199),(106,198),(107,197),(108,196),(109,195),(110,194),(111,193),(112,192),(113,191),(114,190),(115,189),(116,188),(117,187),(118,186),(119,185),(120,184),(121,183),(122,208),(123,207),(124,206),(125,205),(126,204),(127,203),(128,202),(129,201),(130,200)], [(1,39,165,85),(2,38,166,84),(3,37,167,83),(4,36,168,82),(5,35,169,81),(6,34,170,80),(7,33,171,79),(8,32,172,104),(9,31,173,103),(10,30,174,102),(11,29,175,101),(12,28,176,100),(13,27,177,99),(14,52,178,98),(15,51,179,97),(16,50,180,96),(17,49,181,95),(18,48,182,94),(19,47,157,93),(20,46,158,92),(21,45,159,91),(22,44,160,90),(23,43,161,89),(24,42,162,88),(25,41,163,87),(26,40,164,86),(53,109,150,198),(54,108,151,197),(55,107,152,196),(56,106,153,195),(57,105,154,194),(58,130,155,193),(59,129,156,192),(60,128,131,191),(61,127,132,190),(62,126,133,189),(63,125,134,188),(64,124,135,187),(65,123,136,186),(66,122,137,185),(67,121,138,184),(68,120,139,183),(69,119,140,208),(70,118,141,207),(71,117,142,206),(72,116,143,205),(73,115,144,204),(74,114,145,203),(75,113,146,202),(76,112,147,201),(77,111,148,200),(78,110,149,199)], [(1,121,165,184),(2,122,166,185),(3,123,167,186),(4,124,168,187),(5,125,169,188),(6,126,170,189),(7,127,171,190),(8,128,172,191),(9,129,173,192),(10,130,174,193),(11,105,175,194),(12,106,176,195),(13,107,177,196),(14,108,178,197),(15,109,179,198),(16,110,180,199),(17,111,181,200),(18,112,182,201),(19,113,157,202),(20,114,158,203),(21,115,159,204),(22,116,160,205),(23,117,161,206),(24,118,162,207),(25,119,163,208),(26,120,164,183),(27,55,99,152),(28,56,100,153),(29,57,101,154),(30,58,102,155),(31,59,103,156),(32,60,104,131),(33,61,79,132),(34,62,80,133),(35,63,81,134),(36,64,82,135),(37,65,83,136),(38,66,84,137),(39,67,85,138),(40,68,86,139),(41,69,87,140),(42,70,88,141),(43,71,89,142),(44,72,90,143),(45,73,91,144),(46,74,92,145),(47,75,93,146),(48,76,94,147),(49,77,95,148),(50,78,96,149),(51,53,97,150),(52,54,98,151)])

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H13A···13F26A···26R52A···52AJ
order1222224444444413···1326···2652···52
size111126262244262652522···22···24···4

74 irreducible representations

dim11111122222244
type+++++++-+++--
imageC1C2C2C2C2C2D4Q8C4○D4D13D26D52D42D13Q8×D13
kernelD262Q8C523C4D26⋊C4C13×C4⋊C4C2×Dic26C2×C4×D13C52D26C26C4⋊C4C2×C4C4C2C2
# reps1221112226182466

Matrix representation of D262Q8 in GL4(𝔽53) generated by

504700
64700
00520
00052
,
141900
263900
0010
001552
,
132200
24000
005239
00381
,
52000
05200
00300
002623
G:=sub<GL(4,GF(53))| [50,6,0,0,47,47,0,0,0,0,52,0,0,0,0,52],[14,26,0,0,19,39,0,0,0,0,1,15,0,0,0,52],[13,2,0,0,22,40,0,0,0,0,52,38,0,0,39,1],[52,0,0,0,0,52,0,0,0,0,30,26,0,0,0,23] >;

D262Q8 in GAP, Magma, Sage, TeX

D_{26}\rtimes_2Q_8
% in TeX

G:=Group("D26:2Q8");
// GroupNames label

G:=SmallGroup(416,118);
// by ID

G=gap.SmallGroup(416,118);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,122,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽