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G = D26⋊D4order 416 = 25·13

1st semidirect product of D26 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D261D4, Dic132D4, C23.5D26, (C2×D52)⋊3C2, C2.9(D4×D13), (C2×C4).7D26, C131(C4⋊D4), C22⋊C44D13, C26.20(C2×D4), C26.9(C4○D4), C26.D45C2, D26⋊C411C2, (C2×C26).25C23, (C2×C52).53C22, C2.11(D525C2), (C22×C26).14C22, (C2×Dic13).7C22, C22.43(C22×D13), (C22×D13).21C22, (C2×C4×D13)⋊11C2, (C2×C13⋊D4)⋊2C2, (C13×C22⋊C4)⋊6C2, SmallGroup(416,105)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D26⋊D4
C1C13C26C2×C26C22×D13C2×C4×D13 — D26⋊D4
C13C2×C26 — D26⋊D4
C1C22C22⋊C4

Generators and relations for D26⋊D4
 G = < a,b,c,d | a26=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a24b, dbd=a11b, dcd=c-1 >

Subgroups: 760 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22, C22 [×10], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C13, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], D13 [×3], C26 [×3], C26, C4⋊D4, Dic13 [×2], Dic13, C52 [×2], D26 [×2], D26 [×5], C2×C26, C2×C26 [×3], C4×D13 [×2], D52 [×2], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52 [×2], C22×D13 [×2], C22×C26, C26.D4, D26⋊C4, C13×C22⋊C4, C2×C4×D13, C2×D52, C2×C13⋊D4 [×2], D26⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, D13, C4⋊D4, D26 [×3], C22×D13, D525C2, D4×D13 [×2], D26⋊D4

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F13A···13F26A···26R26S···26AD52A···52X
order1222222244444413···1326···2626···2652···52
size111142626522242626522···22···24···44···4

74 irreducible representations

dim111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4D13D26D26D525C2D4×D13
kernelD26⋊D4C26.D4D26⋊C4C13×C22⋊C4C2×C4×D13C2×D52C2×C13⋊D4Dic13D26C26C22⋊C4C2×C4C23C2C2
# reps111111222261262412

Matrix representation of D26⋊D4 in GL4(𝔽53) generated by

173300
162800
0010
0001
,
414400
101200
00520
00052
,
172900
433600
00738
002146
,
411400
391200
004625
00327
G:=sub<GL(4,GF(53))| [17,16,0,0,33,28,0,0,0,0,1,0,0,0,0,1],[41,10,0,0,44,12,0,0,0,0,52,0,0,0,0,52],[17,43,0,0,29,36,0,0,0,0,7,21,0,0,38,46],[41,39,0,0,14,12,0,0,0,0,46,32,0,0,25,7] >;

D26⋊D4 in GAP, Magma, Sage, TeX

D_{26}\rtimes D_4
% in TeX

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

G:=SmallGroup(416,105);
// by ID

G=gap.SmallGroup(416,105);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,506,188,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^24*b,d*b*d=a^11*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽