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G = D83D13order 416 = 25·13

The semidirect product of D8 and D13 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83D13, C8.8D26, D4.1D26, D26.5D4, Dic524C2, C52.3C23, C104.6C22, Dic13.24D4, Dic26.1C22, (C13×D8)⋊3C2, (C8×D13)⋊2C2, C132(C4○D8), D4.D132C2, C2.17(D4×D13), C26.29(C2×D4), D42D132C2, C4.3(C22×D13), C132C8.5C22, (D4×C13).1C22, (C4×D13).16C22, SmallGroup(416,133)

Series: Derived Chief Lower central Upper central

C1C52 — D83D13
C1C13C26C52C4×D13D42D13 — D83D13
C13C26C52 — D83D13
C1C2C4D8

Generators and relations for D83D13
 G = < a,b,c,d | a8=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 416 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C13, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D13, C26, C26 [×2], C4○D8, Dic13, Dic13 [×2], C52, D26, C2×C26 [×2], C132C8, C104, Dic26 [×2], C4×D13, C2×Dic13 [×2], C13⋊D4 [×2], D4×C13 [×2], C8×D13, Dic52, D4.D13 [×2], C13×D8, D42D13 [×2], D83D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], C22×D13, D4×D13, D83D13

Smallest permutation representation of D83D13
On 208 points
Generators in S208
(1 188 27 171 22 207 49 168)(2 189 28 172 23 208 50 169)(3 190 29 173 24 196 51 157)(4 191 30 174 25 197 52 158)(5 192 31 175 26 198 40 159)(6 193 32 176 14 199 41 160)(7 194 33 177 15 200 42 161)(8 195 34 178 16 201 43 162)(9 183 35 179 17 202 44 163)(10 184 36 180 18 203 45 164)(11 185 37 181 19 204 46 165)(12 186 38 182 20 205 47 166)(13 187 39 170 21 206 48 167)(53 152 102 119 75 133 91 107)(54 153 103 120 76 134 79 108)(55 154 104 121 77 135 80 109)(56 155 92 122 78 136 81 110)(57 156 93 123 66 137 82 111)(58 144 94 124 67 138 83 112)(59 145 95 125 68 139 84 113)(60 146 96 126 69 140 85 114)(61 147 97 127 70 141 86 115)(62 148 98 128 71 142 87 116)(63 149 99 129 72 143 88 117)(64 150 100 130 73 131 89 105)(65 151 101 118 74 132 90 106)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 105)(12 106)(13 107)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 141)(35 142)(36 143)(37 131)(38 132)(39 133)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 167)(54 168)(55 169)(56 157)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 170)(76 171)(77 172)(78 173)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 183)(88 184)(89 185)(90 186)(91 187)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 16)(17 26)(18 25)(19 24)(20 23)(21 22)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 44)(41 43)(45 52)(46 51)(47 50)(48 49)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 78)(65 77)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 104)(91 103)(105 122)(106 121)(107 120)(108 119)(109 118)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(131 155)(132 154)(133 153)(134 152)(135 151)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(143 156)(157 165)(158 164)(159 163)(160 162)(166 169)(167 168)(170 171)(172 182)(173 181)(174 180)(175 179)(176 178)(183 192)(184 191)(185 190)(186 189)(187 188)(193 195)(196 204)(197 203)(198 202)(199 201)(205 208)(206 207)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D13A···13F26A···26F26G···26R52A···52F104A···104L
order1222244444888813···1326···2626···2652···52104···104
size1144262131352522226262···22···28···84···44···4

56 irreducible representations

dim11111122222244
type++++++++++++-
imageC1C2C2C2C2C2D4D4D13C4○D8D26D26D4×D13D83D13
kernelD83D13C8×D13Dic52D4.D13C13×D8D42D13Dic13D26D8C13C8D4C2C1
# reps1112121164612612

Matrix representation of D83D13 in GL4(𝔽313) generated by

5000
2518800
003120
000312
,
4924600
20426400
003120
000312
,
1000
0100
002881
0048286
,
1000
19331200
00162261
00294151
G:=sub<GL(4,GF(313))| [5,25,0,0,0,188,0,0,0,0,312,0,0,0,0,312],[49,204,0,0,246,264,0,0,0,0,312,0,0,0,0,312],[1,0,0,0,0,1,0,0,0,0,288,48,0,0,1,286],[1,193,0,0,0,312,0,0,0,0,162,294,0,0,261,151] >;

D83D13 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_{13}
% in TeX

G:=Group("D8:3D13");
// GroupNames label

G:=SmallGroup(416,133);
// by ID

G=gap.SmallGroup(416,133);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,362,116,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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