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G = D4.9D26order 416 = 25·13

4th non-split extension by D4 of D26 acting via D26/C26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D26, C52.50D4, Q8.9D26, C52.19C23, Dic26.12C22, D4.D136C2, C4○D4.2D13, C13⋊Q166C2, C26.61(C2×D4), (C2×C4).23D26, (C2×C26).10D4, C135(C8.C22), C52.4C410C2, (C2×Dic26)⋊11C2, C4.25(C13⋊D4), (C2×C52).44C22, C132C8.4C22, (D4×C13).9C22, C4.19(C22×D13), (Q8×C13).9C22, C22.6(C13⋊D4), (C13×C4○D4).3C2, C2.25(C2×C13⋊D4), SmallGroup(416,172)

Series: Derived Chief Lower central Upper central

C1C52 — D4.9D26
C1C13C26C52Dic26C2×Dic26 — D4.9D26
C13C26C52 — D4.9D26
C1C2C2×C4C4○D4

Generators and relations for D4.9D26
 G = < a,b,c,d | a4=b2=c26=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 312 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C13, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C26, C26 [×2], C8.C22, Dic13 [×2], C52 [×2], C52, C2×C26, C2×C26, C132C8 [×2], Dic26 [×2], Dic26, C2×Dic13, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C52.4C4, D4.D13 [×2], C13⋊Q16 [×2], C2×Dic26, C13×C4○D4, D4.9D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8.C22, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, D4.9D26

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

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

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

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

71 conjugacy classes

class 1 2A2B2C4A4B4C4D4E8A8B13A···13F26A···26F26G···26X52A···52L52M···52AD
order1222444448813···1326···2626···2652···5252···52
size1124224525252522···22···24···42···24···4

71 irreducible representations

dim1111112222222244
type++++++++++++--
imageC1C2C2C2C2C2D4D4D13D26D26D26C13⋊D4C13⋊D4C8.C22D4.9D26
kernelD4.9D26C52.4C4D4.D13C13⋊Q16C2×Dic26C13×C4○D4C52C2×C26C4○D4C2×C4D4Q8C4C22C13C1
# reps1122111166661212112

Matrix representation of D4.9D26 in GL4(𝔽313) generated by

15420200
515900
6161227111
56625286
,
79301150139
010386146
31009112
11416017340
,
19826400
115000
2719111449
98171621
,
61100
8230700
26132414
192222200309
G:=sub<GL(4,GF(313))| [154,5,61,5,202,159,61,66,0,0,227,252,0,0,111,86],[79,0,3,114,301,103,100,160,150,86,91,173,139,146,12,40],[198,115,27,98,264,0,191,17,0,0,114,162,0,0,49,1],[6,82,26,192,11,307,132,222,0,0,4,200,0,0,14,309] >;

D4.9D26 in GAP, Magma, Sage, TeX

D_4._9D_{26}
% in TeX

G:=Group("D4.9D26");
// GroupNames label

G:=SmallGroup(416,172);
// by ID

G=gap.SmallGroup(416,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,218,188,579,159,69,13829]);
// Polycyclic

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

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