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## G = Q8.D26order 416 = 25·13

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Q8.D26
 Chief series C1 — C13 — C26 — C52 — D52 — D52⋊5C2 — Q8.D26
 Lower central C13 — C26 — C52 — Q8.D26
 Upper central C1 — C2 — C2×C4 — C2×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, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C13, M4(2), SD16, Q16, C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13, C52, C52, D26, C2×C26, C132C8, Dic26, C4×D13, D52, C13⋊D4, C2×C52, C2×C52, Q8×C13, Q8×C13, C52.4C4, Q8⋊D13, C13⋊Q16, D525C2, Q8×C26, Q8.D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8.C22, D26, C13⋊D4, 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)(105 208 131 182)(106 183 132 157)(107 158 133 184)(108 185 134 159)(109 160 135 186)(110 187 136 161)(111 162 137 188)(112 189 138 163)(113 164 139 190)(114 191 140 165)(115 166 141 192)(116 193 142 167)(117 168 143 194)(118 195 144 169)(119 170 145 196)(120 197 146 171)(121 172 147 198)(122 199 148 173)(123 174 149 200)(124 201 150 175)(125 176 151 202)(126 203 152 177)(127 178 153 204)(128 205 154 179)(129 180 155 206)(130 207 156 181)
(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)
(1 118 27 144)(2 143 28 117)(3 116 29 142)(4 141 30 115)(5 114 31 140)(6 139 32 113)(7 112 33 138)(8 137 34 111)(9 110 35 136)(10 135 36 109)(11 108 37 134)(12 133 38 107)(13 106 39 132)(14 131 40 105)(15 156 41 130)(16 129 42 155)(17 154 43 128)(18 127 44 153)(19 152 45 126)(20 125 46 151)(21 150 47 124)(22 123 48 149)(23 148 49 122)(24 121 50 147)(25 146 51 120)(26 119 52 145)(53 196 79 170)(54 169 80 195)(55 194 81 168)(56 167 82 193)(57 192 83 166)(58 165 84 191)(59 190 85 164)(60 163 86 189)(61 188 87 162)(62 161 88 187)(63 186 89 160)(64 159 90 185)(65 184 91 158)(66 157 92 183)(67 182 93 208)(68 207 94 181)(69 180 95 206)(70 205 96 179)(71 178 97 204)(72 203 98 177)(73 176 99 202)(74 201 100 175)(75 174 101 200)(76 199 102 173)(77 172 103 198)(78 197 104 171)

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

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

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

71 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 2 52 2 2 4 4 52 52 52 2 ··· 2 2 ··· 2 4 ··· 4

71 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 C13⋊D4 C13⋊D4 C8.C22 Q8.D26 kernel Q8.D26 C52.4C4 Q8⋊D13 C13⋊Q16 D52⋊5C2 Q8×C26 C52 C2×C26 C2×Q8 C2×C4 Q8 C4 C22 C13 C1 # reps 1 1 2 2 1 1 1 1 6 6 12 12 12 1 12

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

 312 192 211 26 238 1 158 254 0 0 1 48 0 0 13 312
,
 5 229 162 295 127 308 132 302 0 0 278 219 0 0 53 35
,
 249 81 264 282 208 64 251 135 0 0 44 234 0 0 259 269
,
 210 57 211 18 20 233 282 116 58 132 179 121 128 232 268 4
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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