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G = D4.10D26order 416 = 25·13

The non-split extension by D4 of D26 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D26, Q8.11D26, C52.27C23, C26.13C24, D26.8C23, C1322- 1+4, D52.14C22, Dic13.8C23, Dic26.14C22, C4○D44D13, (Q8×D13)⋊5C2, (C2×C4).25D26, C13⋊D4.C22, D42D135C2, D525C29C2, (C2×C26).5C23, (C2×Dic26)⋊14C2, (C2×C52).49C22, (C4×D13).6C22, C2.14(C23×D13), C4.34(C22×D13), (D4×C13).10C22, (Q8×C13).11C22, C22.4(C22×D13), (C2×Dic13).22C22, (C13×C4○D4)⋊5C2, SmallGroup(416,224)

Series: Derived Chief Lower central Upper central

C1C26 — D4.10D26
C1C13C26D26C4×D13Q8×D13 — D4.10D26
C13C26 — D4.10D26
C1C2C4○D4

Generators and relations for D4.10D26
 G = < a,b,c,d | a4=b2=1, c26=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c25 >

Subgroups: 816 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C13, C2×Q8 [×5], C4○D4, C4○D4 [×9], D13 [×2], C26, C26 [×3], 2- 1+4, Dic13 [×6], C52, C52 [×3], D26 [×2], C2×C26 [×3], Dic26 [×9], C4×D13 [×6], D52, C2×Dic13 [×6], C13⋊D4 [×6], C2×C52 [×3], D4×C13 [×3], Q8×C13, C2×Dic26 [×3], D525C2 [×3], D42D13 [×6], Q8×D13 [×2], C13×C4○D4, D4.10D26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, D13, 2- 1+4, D26 [×7], C22×D13 [×7], C23×D13, D4.10D26

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

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

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

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

77 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J13A···13F26A···26F26G···26X52A···52L52M···52AD
order122222244444···413···1326···2626···2652···5252···52
size112222626222226···262···22···24···42···24···4

77 irreducible representations

dim111111222244
type++++++++++--
imageC1C2C2C2C2C2D13D26D26D262- 1+4D4.10D26
kernelD4.10D26C2×Dic26D525C2D42D13Q8×D13C13×C4○D4C4○D4C2×C4D4Q8C13C1
# reps133621618186112

Matrix representation of D4.10D26 in GL4(𝔽53) generated by

280118
0285229
4030250
1616025
,
2149283
33325215
93034
43382250
,
31462039
18374310
245117
1191227
,
48473434
451911
41135234
404001
G:=sub<GL(4,GF(53))| [28,0,40,16,0,28,30,16,1,52,25,0,18,29,0,25],[21,33,9,43,49,32,30,38,28,52,3,22,3,15,4,50],[31,18,2,1,46,37,45,19,20,43,11,12,39,10,7,27],[48,4,41,40,47,5,13,40,34,19,52,0,34,11,34,1] >;

D4.10D26 in GAP, Magma, Sage, TeX

D_4._{10}D_{26}
% in TeX

G:=Group("D4.10D26");
// GroupNames label

G:=SmallGroup(416,224);
// by ID

G=gap.SmallGroup(416,224);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,188,86,579,69,13829]);
// 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^2*b,b*d=d*b,d*c*d^-1=c^25>;
// generators/relations

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