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G = Q8.D26order 416 = 25·13

1st non-split extension by Q8 of D26 acting via D26/C26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.19D4, Q8.6D26, C52.15C23, D52.10C22, Dic26.9C22, Q8⋊D135C2, (C2×Q8)⋊2D13, (Q8×C26)⋊2C2, C13⋊Q165C2, C26.54(C2×D4), (C2×C26).42D4, (C2×C4).20D26, C52.4C47C2, C134(C8.C22), D525C2.5C2, C4.17(C13⋊D4), (C2×C52).37C22, C132C8.3C22, C4.15(C22×D13), (Q8×C13).6C22, C22.11(C13⋊D4), C2.18(C2×C13⋊D4), SmallGroup(416,163)

Series: Derived Chief Lower central Upper central

C1C52 — Q8.D26
C1C13C26C52D52D525C2 — Q8.D26
C13C26C52 — Q8.D26
C1C2C2×C4C2×Q8

Generators and relations for Q8.D26
 G = < a,b,c,d | a4=1, b2=c26=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c25 >

Subgroups: 360 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], C13, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13, C52 [×2], C52 [×2], D26, C2×C26, C132C8 [×2], Dic26, C4×D13, D52, C13⋊D4, C2×C52, C2×C52, Q8×C13 [×2], Q8×C13, C52.4C4, Q8⋊D13 [×2], C13⋊Q16 [×2], D525C2, Q8×C26, Q8.D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8.C22, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, Q8.D26

Smallest permutation representation of Q8.D26
On 208 points
Generators in S208
(1 40 27 14)(2 41 28 15)(3 42 29 16)(4 43 30 17)(5 44 31 18)(6 45 32 19)(7 46 33 20)(8 47 34 21)(9 48 35 22)(10 49 36 23)(11 50 37 24)(12 51 38 25)(13 52 39 26)(53 92 79 66)(54 93 80 67)(55 94 81 68)(56 95 82 69)(57 96 83 70)(58 97 84 71)(59 98 85 72)(60 99 86 73)(61 100 87 74)(62 101 88 75)(63 102 89 76)(64 103 90 77)(65 104 91 78)(105 118 131 144)(106 119 132 145)(107 120 133 146)(108 121 134 147)(109 122 135 148)(110 123 136 149)(111 124 137 150)(112 125 138 151)(113 126 139 152)(114 127 140 153)(115 128 141 154)(116 129 142 155)(117 130 143 156)(157 170 183 196)(158 171 184 197)(159 172 185 198)(160 173 186 199)(161 174 187 200)(162 175 188 201)(163 176 189 202)(164 177 190 203)(165 178 191 204)(166 179 192 205)(167 180 193 206)(168 181 194 207)(169 182 195 208)
(1 67 27 93)(2 94 28 68)(3 69 29 95)(4 96 30 70)(5 71 31 97)(6 98 32 72)(7 73 33 99)(8 100 34 74)(9 75 35 101)(10 102 36 76)(11 77 37 103)(12 104 38 78)(13 79 39 53)(14 54 40 80)(15 81 41 55)(16 56 42 82)(17 83 43 57)(18 58 44 84)(19 85 45 59)(20 60 46 86)(21 87 47 61)(22 62 48 88)(23 89 49 63)(24 64 50 90)(25 91 51 65)(26 66 52 92)(105 179 131 205)(106 206 132 180)(107 181 133 207)(108 208 134 182)(109 183 135 157)(110 158 136 184)(111 185 137 159)(112 160 138 186)(113 187 139 161)(114 162 140 188)(115 189 141 163)(116 164 142 190)(117 191 143 165)(118 166 144 192)(119 193 145 167)(120 168 146 194)(121 195 147 169)(122 170 148 196)(123 197 149 171)(124 172 150 198)(125 199 151 173)(126 174 152 200)(127 201 153 175)(128 176 154 202)(129 203 155 177)(130 178 156 204)
(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)
(1 113 27 139)(2 138 28 112)(3 111 29 137)(4 136 30 110)(5 109 31 135)(6 134 32 108)(7 107 33 133)(8 132 34 106)(9 105 35 131)(10 130 36 156)(11 155 37 129)(12 128 38 154)(13 153 39 127)(14 126 40 152)(15 151 41 125)(16 124 42 150)(17 149 43 123)(18 122 44 148)(19 147 45 121)(20 120 46 146)(21 145 47 119)(22 118 48 144)(23 143 49 117)(24 116 50 142)(25 141 51 115)(26 114 52 140)(53 188 79 162)(54 161 80 187)(55 186 81 160)(56 159 82 185)(57 184 83 158)(58 157 84 183)(59 182 85 208)(60 207 86 181)(61 180 87 206)(62 205 88 179)(63 178 89 204)(64 203 90 177)(65 176 91 202)(66 201 92 175)(67 174 93 200)(68 199 94 173)(69 172 95 198)(70 197 96 171)(71 170 97 196)(72 195 98 169)(73 168 99 194)(74 193 100 167)(75 166 101 192)(76 191 102 165)(77 164 103 190)(78 189 104 163)

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

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

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

71 conjugacy classes

class 1 2A2B2C4A4B4C4D4E8A8B13A···13F26A···26R52A···52AJ
order1222444448813···1326···2652···52
size1125222445252522···22···24···4

71 irreducible representations

dim111111222222244
type+++++++++++-
imageC1C2C2C2C2C2D4D4D13D26D26C13⋊D4C13⋊D4C8.C22Q8.D26
kernelQ8.D26C52.4C4Q8⋊D13C13⋊Q16D525C2Q8×C26C52C2×C26C2×Q8C2×C4Q8C4C22C13C1
# reps1122111166121212112

Matrix representation of Q8.D26 in GL4(𝔽313) generated by

31219221126
2381158254
00148
0013312
,
5229162295
127308132302
00278219
005335
,
24981264282
20864251135
0044234
00259269
,
2105721118
20233282116
58132179121
1282322684
G:=sub<GL(4,GF(313))| [312,238,0,0,192,1,0,0,211,158,1,13,26,254,48,312],[5,127,0,0,229,308,0,0,162,132,278,53,295,302,219,35],[249,208,0,0,81,64,0,0,264,251,44,259,282,135,234,269],[210,20,58,128,57,233,132,232,211,282,179,268,18,116,121,4] >;

Q8.D26 in GAP, Magma, Sage, TeX

Q_8.D_{26}
% in TeX

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

G:=SmallGroup(416,163);
// by ID

G=gap.SmallGroup(416,163);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,86,579,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=1,b^2=c^26=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^25>;
// generators/relations

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