Copied to
clipboard

G = D4.D26order 416 = 25·13

4th non-split extension by D4 of D26 acting via D26/D13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2D26, D4.4D26, Q8.1D26, Dic526C2, SD162D13, D26.16D4, C52.6C23, C104.9C22, Dic13.18D4, Dic26.2C22, (Q8×D13)⋊2C2, C8⋊D132C2, D4.D134C2, C13⋊Q161C2, C26.32(C2×D4), C2.20(D4×D13), (C13×SD16)⋊2C2, C132(C8.C22), C4.6(C22×D13), D42D13.1C2, C132C8.1C22, (D4×C13).4C22, (C4×D13).3C22, (Q8×C13).1C22, SmallGroup(416,136)

Series: Derived Chief Lower central Upper central

C1C52 — D4.D26
C1C13C26C52C4×D13Q8×D13 — D4.D26
C13C26C52 — D4.D26
C1C2C4SD16

Generators and relations for D4.D26
 G = < a,b,c,d | a4=b2=1, c26=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c25 >

Subgroups: 408 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C13, M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13, Dic13 [×2], C52, C52, D26, C2×C26, C132C8, C104, Dic26 [×2], Dic26, C4×D13, C4×D13, C2×Dic13, C13⋊D4, D4×C13, Q8×C13, C8⋊D13, Dic52, D4.D13, C13⋊Q16, C13×SD16, D42D13, Q8×D13, D4.D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8.C22, D26 [×3], C22×D13, D4×D13, D4.D26

Smallest permutation representation of D4.D26
On 208 points
Generators in S208
(1 191 27 165)(2 166 28 192)(3 193 29 167)(4 168 30 194)(5 195 31 169)(6 170 32 196)(7 197 33 171)(8 172 34 198)(9 199 35 173)(10 174 36 200)(11 201 37 175)(12 176 38 202)(13 203 39 177)(14 178 40 204)(15 205 41 179)(16 180 42 206)(17 207 43 181)(18 182 44 208)(19 157 45 183)(20 184 46 158)(21 159 47 185)(22 186 48 160)(23 161 49 187)(24 188 50 162)(25 163 51 189)(26 190 52 164)(53 114 79 140)(54 141 80 115)(55 116 81 142)(56 143 82 117)(57 118 83 144)(58 145 84 119)(59 120 85 146)(60 147 86 121)(61 122 87 148)(62 149 88 123)(63 124 89 150)(64 151 90 125)(65 126 91 152)(66 153 92 127)(67 128 93 154)(68 155 94 129)(69 130 95 156)(70 105 96 131)(71 132 97 106)(72 107 98 133)(73 134 99 108)(74 109 100 135)(75 136 101 110)(76 111 102 137)(77 138 103 112)(78 113 104 139)
(1 151)(2 65)(3 153)(4 67)(5 155)(6 69)(7 105)(8 71)(9 107)(10 73)(11 109)(12 75)(13 111)(14 77)(15 113)(16 79)(17 115)(18 81)(19 117)(20 83)(21 119)(22 85)(23 121)(24 87)(25 123)(26 89)(27 125)(28 91)(29 127)(30 93)(31 129)(32 95)(33 131)(34 97)(35 133)(36 99)(37 135)(38 101)(39 137)(40 103)(41 139)(42 53)(43 141)(44 55)(45 143)(46 57)(47 145)(48 59)(49 147)(50 61)(51 149)(52 63)(54 181)(56 183)(58 185)(60 187)(62 189)(64 191)(66 193)(68 195)(70 197)(72 199)(74 201)(76 203)(78 205)(80 207)(82 157)(84 159)(86 161)(88 163)(90 165)(92 167)(94 169)(96 171)(98 173)(100 175)(102 177)(104 179)(106 172)(108 174)(110 176)(112 178)(114 180)(116 182)(118 184)(120 186)(122 188)(124 190)(126 192)(128 194)(130 196)(132 198)(134 200)(136 202)(138 204)(140 206)(142 208)(144 158)(146 160)(148 162)(150 164)(152 166)(154 168)(156 170)
(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 26 27 52)(2 51 28 25)(3 24 29 50)(4 49 30 23)(5 22 31 48)(6 47 32 21)(7 20 33 46)(8 45 34 19)(9 18 35 44)(10 43 36 17)(11 16 37 42)(12 41 38 15)(13 14 39 40)(53 100 79 74)(54 73 80 99)(55 98 81 72)(56 71 82 97)(57 96 83 70)(58 69 84 95)(59 94 85 68)(60 67 86 93)(61 92 87 66)(62 65 88 91)(63 90 89 64)(75 78 101 104)(76 103 102 77)(105 144 131 118)(106 117 132 143)(107 142 133 116)(108 115 134 141)(109 140 135 114)(110 113 136 139)(111 138 137 112)(119 130 145 156)(120 155 146 129)(121 128 147 154)(122 153 148 127)(123 126 149 152)(124 151 150 125)(157 198 183 172)(158 171 184 197)(159 196 185 170)(160 169 186 195)(161 194 187 168)(162 167 188 193)(163 192 189 166)(164 165 190 191)(173 182 199 208)(174 207 200 181)(175 180 201 206)(176 205 202 179)(177 178 203 204)

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

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

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

53 conjugacy classes

class 1 2A2B2C4A4B4C4D4E8A8B13A···13F26A···26F26G···26L52A···52F52G···52L104A···104L
order1222444448813···1326···2626···2652···5252···52104···104
size11426242652524522···22···28···84···48···84···4

53 irreducible representations

dim11111111222222444
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4D13D26D26D26C8.C22D4×D13D4.D26
kernelD4.D26C8⋊D13Dic52D4.D13C13⋊Q16C13×SD16D42D13Q8×D13Dic13D26SD16C8D4Q8C13C2C1
# reps111111111166661612

Matrix representation of D4.D26 in GL4(𝔽313) generated by

27923351201
221551114
14715413580
23816126157
,
29068230119
14634257143
42159274245
16517323628
,
251252129123
14217277189
3112141261
5019455191
,
8682123129
9211118977
2502466112
21619119155
G:=sub<GL(4,GF(313))| [279,221,147,238,233,55,154,16,51,111,135,126,201,4,80,157],[290,146,42,165,68,34,159,173,230,257,274,236,119,143,245,28],[251,142,311,50,252,172,214,194,129,77,12,55,123,189,61,191],[86,92,250,216,82,111,246,191,123,189,61,191,129,77,12,55] >;

D4.D26 in GAP, Magma, Sage, TeX

D_4.D_{26}
% in TeX

G:=Group("D4.D26");
// GroupNames label

G:=SmallGroup(416,136);
// by ID

G=gap.SmallGroup(416,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,116,297,159,69,13829]);
// Polycyclic

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

׿
×
𝔽