Copied to
clipboard

G = D8.D13order 416 = 25·13

The non-split extension by D8 of D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.D13, C26.9D8, C8.5D26, C52.4D4, C132SD32, Dic523C2, C104.3C22, C132C162C2, (C13×D8).1C2, C2.5(D4⋊D13), C4.2(C13⋊D4), SmallGroup(416,34)

Series: Derived Chief Lower central Upper central

C1C104 — D8.D13
C1C13C26C52C104Dic52 — D8.D13
C13C26C52C104 — D8.D13
C1C2C4C8D8

Generators and relations for D8.D13
 G = < a,b,c,d | a8=b2=c13=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

8C2
4C22
52C4
8C26
2D4
26Q8
4Dic13
4C2×C26
13C16
13Q16
2Dic26
2D4×C13
13SD32

Smallest permutation representation of D8.D13
On 208 points
Generators in S208
(1 87 29 71 15 96 43 54)(2 88 30 72 16 97 44 55)(3 89 31 73 17 98 45 56)(4 90 32 74 18 99 46 57)(5 91 33 75 19 100 47 58)(6 79 34 76 20 101 48 59)(7 80 35 77 21 102 49 60)(8 81 36 78 22 103 50 61)(9 82 37 66 23 104 51 62)(10 83 38 67 24 92 52 63)(11 84 39 68 25 93 40 64)(12 85 27 69 26 94 41 65)(13 86 28 70 14 95 42 53)(105 157 144 196 118 170 131 183)(106 158 145 197 119 171 132 184)(107 159 146 198 120 172 133 185)(108 160 147 199 121 173 134 186)(109 161 148 200 122 174 135 187)(110 162 149 201 123 175 136 188)(111 163 150 202 124 176 137 189)(112 164 151 203 125 177 138 190)(113 165 152 204 126 178 139 191)(114 166 153 205 127 179 140 192)(115 167 154 206 128 180 141 193)(116 168 155 207 129 181 142 194)(117 169 156 208 130 182 143 195)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 53)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 66)(24 67)(25 68)(26 69)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 92)(39 93)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 79)(49 80)(50 81)(51 82)(52 83)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(157 170)(158 171)(159 172)(160 173)(161 174)(162 175)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 182)
(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)
(1 130 15 117)(2 129 16 116)(3 128 17 115)(4 127 18 114)(5 126 19 113)(6 125 20 112)(7 124 21 111)(8 123 22 110)(9 122 23 109)(10 121 24 108)(11 120 25 107)(12 119 26 106)(13 118 14 105)(27 145 41 132)(28 144 42 131)(29 156 43 143)(30 155 44 142)(31 154 45 141)(32 153 46 140)(33 152 47 139)(34 151 48 138)(35 150 49 137)(36 149 50 136)(37 148 51 135)(38 147 52 134)(39 146 40 133)(53 170 70 157)(54 182 71 169)(55 181 72 168)(56 180 73 167)(57 179 74 166)(58 178 75 165)(59 177 76 164)(60 176 77 163)(61 175 78 162)(62 174 66 161)(63 173 67 160)(64 172 68 159)(65 171 69 158)(79 203 101 190)(80 202 102 189)(81 201 103 188)(82 200 104 187)(83 199 92 186)(84 198 93 185)(85 197 94 184)(86 196 95 183)(87 208 96 195)(88 207 97 194)(89 206 98 193)(90 205 99 192)(91 204 100 191)

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

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

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

53 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F16A16B16C16D26A···26F26G···26R52A···52F104A···104L
order122448813···131616161626···2626···2652···52104···104
size1182104222···2262626262···28···84···44···4

53 irreducible representations

dim111122222244
type+++++++++-
imageC1C2C2C2D4D8D13SD32D26C13⋊D4D4⋊D13D8.D13
kernelD8.D13C132C16Dic52C13×D8C52C26D8C13C8C4C2C1
# reps11111264612612

Matrix representation of D8.D13 in GL4(𝔽1249) generated by

1248000
0124800
000142
0081850
,
1248000
396100
000142
004310
,
83000
93493300
0010
0001
,
31631700
93493300
0011881027
00118761
G:=sub<GL(4,GF(1249))| [1248,0,0,0,0,1248,0,0,0,0,0,818,0,0,142,50],[1248,396,0,0,0,1,0,0,0,0,0,431,0,0,142,0],[83,934,0,0,0,933,0,0,0,0,1,0,0,0,0,1],[316,934,0,0,317,933,0,0,0,0,1188,1187,0,0,1027,61] >;

D8.D13 in GAP, Magma, Sage, TeX

D_8.D_{13}
% in TeX

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

G:=SmallGroup(416,34);
// by ID

G=gap.SmallGroup(416,34);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,73,218,116,122,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^13=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D8.D13 in TeX

׿
×
𝔽