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G = D26.13D4order 416 = 25·13

2nd non-split extension by D26 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.13D4, C4⋊C42D13, (C2×D52).4C2, C26.25(C2×D4), C2.12(D4×D13), (C2×C4).11D26, C26.D46C2, D26⋊C413C2, C26.12(C4○D4), (C2×C26).35C23, (C2×C52).57C22, C2.5(D52⋊C2), C133(C22.D4), C2.14(D525C2), (C22×D13).6C22, C22.49(C22×D13), (C2×Dic13).11C22, (C13×C4⋊C4)⋊5C2, (C2×C4×D13)⋊13C2, SmallGroup(416,115)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D26.13D4
C1C13C26C2×C26C22×D13C2×C4×D13 — D26.13D4
C13C2×C26 — D26.13D4
C1C22C4⋊C4

Generators and relations for D26.13D4
 G = < a,b,c,d | a26=b2=c4=1, d2=a13, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a13b, dcd-1=c-1 >

Subgroups: 632 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22, C22 [×7], C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], C13, C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, D13 [×3], C26 [×3], C22.D4, Dic13 [×2], C52 [×3], D26 [×2], D26 [×5], C2×C26, C4×D13 [×2], D52 [×2], C2×Dic13 [×2], C2×C52 [×3], C22×D13 [×2], C26.D4, D26⋊C4 [×3], C13×C4⋊C4, C2×C4×D13, C2×D52, D26.13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C22.D4, D26 [×3], C22×D13, D525C2, D4×D13, D52⋊C2, D26.13D4

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G13A···13F26A···26R52A···52AJ
order1222222444444413···1326···2652···52
size111126265222442626522···22···24···4

74 irreducible representations

dim1111112222244
type+++++++++++
imageC1C2C2C2C2C2D4C4○D4D13D26D525C2D4×D13D52⋊C2
kernelD26.13D4C26.D4D26⋊C4C13×C4⋊C4C2×C4×D13C2×D52D26C26C4⋊C4C2×C4C2C2C2
# reps113111246182466

Matrix representation of D26.13D4 in GL4(𝔽53) generated by

334700
281100
00520
00052
,
335200
282000
00143
00052
,
201800
223300
00181
004635
,
364300
291700
001029
00243
G:=sub<GL(4,GF(53))| [33,28,0,0,47,11,0,0,0,0,52,0,0,0,0,52],[33,28,0,0,52,20,0,0,0,0,1,0,0,0,43,52],[20,22,0,0,18,33,0,0,0,0,18,46,0,0,1,35],[36,29,0,0,43,17,0,0,0,0,10,2,0,0,29,43] >;

D26.13D4 in GAP, Magma, Sage, TeX

D_{26}._{13}D_4
% in TeX

G:=Group("D26.13D4");
// GroupNames label

G:=SmallGroup(416,115);
// by ID

G=gap.SmallGroup(416,115);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,218,188,86,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^13*b,d*c*d^-1=c^-1>;
// generators/relations

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