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G = D8.10D10order 320 = 26·5

The non-split extension by D8 of D10 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.10D10, D20.47D4, C20.18C24, C40.39C23, Q16.12D10, SD16.2D10, Dic10.47D4, Dic10.12C23, Dic20.16C22, C4○D86D5, C53(Q8○D8), (D5×Q16)⋊7C2, C5⋊D4.3D4, D83D57C2, C4.145(D4×D5), D4.D5.C22, C4○D4.13D10, D10.54(C2×D4), C20.351(C2×D4), (C2×C8).106D10, SD16⋊D56C2, C52C8.9C23, (C8×D5).8C22, C4.18(C23×D5), C8.18(C22×D5), C22.10(D4×D5), (C2×Dic20)⋊23C2, D20.3C48C2, D4.9D108C2, (Q8×D5).2C22, Dic5.60(C2×D4), (C4×D5).11C23, (C5×D4).12C23, D4.12(C22×D5), (C5×D8).10C22, C8⋊D5.2C22, D4.10D106C2, (C5×Q8).12C23, Q8.12(C22×D5), C5⋊Q16.2C22, (C2×C40).106C22, (C2×C20).535C23, C4○D20.56C22, D42D5.2C22, C10.119(C22×D4), (C5×Q16).12C22, (C5×SD16).2C22, C4.Dic5.49C22, (C2×Dic10).206C22, C2.92(C2×D4×D5), (C5×C4○D8)⋊6C2, (C2×C10).15(C2×D4), (C5×C4○D4).23C22, (C2×C4).234(C22×D5), SmallGroup(320,1443)

Series: Derived Chief Lower central Upper central

C1C20 — D8.10D10
C1C5C10C20C4×D5C4○D20D4.10D10 — D8.10D10
C5C10C20 — D8.10D10
C1C2C2×C4C4○D8

Generators and relations for D8.10D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >

Subgroups: 854 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×2], D4 [×9], Q8 [×2], Q8 [×11], D5 [×2], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×2], SD16 [×4], Q16, Q16 [×8], C2×Q8 [×8], C4○D4 [×2], C4○D4 [×11], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], C8○D4, C2×Q16 [×3], C4○D8, C4○D8 [×2], C8.C22 [×6], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], Dic20 [×4], C4.Dic5, D4.D5 [×4], C5⋊Q16 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×Dic10 [×2], C2×Dic10 [×2], C4○D20, C4○D20 [×2], D42D5 [×4], D42D5 [×4], Q8×D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×Dic20, D83D5 [×2], SD16⋊D5 [×4], D5×Q16 [×2], D4.9D10 [×2], C5×C4○D8, D4.10D10 [×2], D8.10D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D8.10D10

Smallest permutation representation of D8.10D10
On 160 points
Generators in S160
(1 95 30 153 123 86 60 13)(2 96 21 154 124 87 51 14)(3 97 22 155 125 88 52 15)(4 98 23 156 126 89 53 16)(5 99 24 157 127 90 54 17)(6 100 25 158 128 81 55 18)(7 91 26 159 129 82 56 19)(8 92 27 160 130 83 57 20)(9 93 28 151 121 84 58 11)(10 94 29 152 122 85 59 12)(31 144 104 41 61 114 134 71)(32 145 105 42 62 115 135 72)(33 146 106 43 63 116 136 73)(34 147 107 44 64 117 137 74)(35 148 108 45 65 118 138 75)(36 149 109 46 66 119 139 76)(37 150 110 47 67 120 140 77)(38 141 101 48 68 111 131 78)(39 142 102 49 69 112 132 79)(40 143 103 50 70 113 133 80)
(1 148)(2 119)(3 150)(4 111)(5 142)(6 113)(7 144)(8 115)(9 146)(10 117)(11 106)(12 137)(13 108)(14 139)(15 110)(16 131)(17 102)(18 133)(19 104)(20 135)(21 46)(22 77)(23 48)(24 79)(25 50)(26 71)(27 42)(28 73)(29 44)(30 75)(31 91)(32 83)(33 93)(34 85)(35 95)(36 87)(37 97)(38 89)(39 99)(40 81)(41 56)(43 58)(45 60)(47 52)(49 54)(51 76)(53 78)(55 80)(57 72)(59 74)(61 82)(62 92)(63 84)(64 94)(65 86)(66 96)(67 88)(68 98)(69 90)(70 100)(101 156)(103 158)(105 160)(107 152)(109 154)(112 127)(114 129)(116 121)(118 123)(120 125)(122 147)(124 149)(126 141)(128 143)(130 145)(132 157)(134 159)(136 151)(138 153)(140 155)
(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)
(1 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 14 151 154)(12 153 152 13)(15 20 155 160)(16 159 156 19)(17 18 157 158)(21 58 51 28)(22 27 52 57)(23 56 53 26)(24 25 54 55)(29 60 59 30)(31 68 61 38)(32 37 62 67)(33 66 63 36)(34 35 64 65)(39 70 69 40)(41 78 71 48)(42 47 72 77)(43 76 73 46)(44 45 74 75)(49 80 79 50)(81 99 100 90)(82 89 91 98)(83 97 92 88)(84 87 93 96)(85 95 94 86)(101 104 131 134)(102 133 132 103)(105 110 135 140)(106 139 136 109)(107 108 137 138)(111 114 141 144)(112 143 142 113)(115 120 145 150)(116 149 146 119)(117 118 147 148)

G:=sub<Sym(160)| (1,95,30,153,123,86,60,13)(2,96,21,154,124,87,51,14)(3,97,22,155,125,88,52,15)(4,98,23,156,126,89,53,16)(5,99,24,157,127,90,54,17)(6,100,25,158,128,81,55,18)(7,91,26,159,129,82,56,19)(8,92,27,160,130,83,57,20)(9,93,28,151,121,84,58,11)(10,94,29,152,122,85,59,12)(31,144,104,41,61,114,134,71)(32,145,105,42,62,115,135,72)(33,146,106,43,63,116,136,73)(34,147,107,44,64,117,137,74)(35,148,108,45,65,118,138,75)(36,149,109,46,66,119,139,76)(37,150,110,47,67,120,140,77)(38,141,101,48,68,111,131,78)(39,142,102,49,69,112,132,79)(40,143,103,50,70,113,133,80), (1,148)(2,119)(3,150)(4,111)(5,142)(6,113)(7,144)(8,115)(9,146)(10,117)(11,106)(12,137)(13,108)(14,139)(15,110)(16,131)(17,102)(18,133)(19,104)(20,135)(21,46)(22,77)(23,48)(24,79)(25,50)(26,71)(27,42)(28,73)(29,44)(30,75)(31,91)(32,83)(33,93)(34,85)(35,95)(36,87)(37,97)(38,89)(39,99)(40,81)(41,56)(43,58)(45,60)(47,52)(49,54)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(62,92)(63,84)(64,94)(65,86)(66,96)(67,88)(68,98)(69,90)(70,100)(101,156)(103,158)(105,160)(107,152)(109,154)(112,127)(114,129)(116,121)(118,123)(120,125)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,14,151,154)(12,153,152,13)(15,20,155,160)(16,159,156,19)(17,18,157,158)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,68,61,38)(32,37,62,67)(33,66,63,36)(34,35,64,65)(39,70,69,40)(41,78,71,48)(42,47,72,77)(43,76,73,46)(44,45,74,75)(49,80,79,50)(81,99,100,90)(82,89,91,98)(83,97,92,88)(84,87,93,96)(85,95,94,86)(101,104,131,134)(102,133,132,103)(105,110,135,140)(106,139,136,109)(107,108,137,138)(111,114,141,144)(112,143,142,113)(115,120,145,150)(116,149,146,119)(117,118,147,148)>;

G:=Group( (1,95,30,153,123,86,60,13)(2,96,21,154,124,87,51,14)(3,97,22,155,125,88,52,15)(4,98,23,156,126,89,53,16)(5,99,24,157,127,90,54,17)(6,100,25,158,128,81,55,18)(7,91,26,159,129,82,56,19)(8,92,27,160,130,83,57,20)(9,93,28,151,121,84,58,11)(10,94,29,152,122,85,59,12)(31,144,104,41,61,114,134,71)(32,145,105,42,62,115,135,72)(33,146,106,43,63,116,136,73)(34,147,107,44,64,117,137,74)(35,148,108,45,65,118,138,75)(36,149,109,46,66,119,139,76)(37,150,110,47,67,120,140,77)(38,141,101,48,68,111,131,78)(39,142,102,49,69,112,132,79)(40,143,103,50,70,113,133,80), (1,148)(2,119)(3,150)(4,111)(5,142)(6,113)(7,144)(8,115)(9,146)(10,117)(11,106)(12,137)(13,108)(14,139)(15,110)(16,131)(17,102)(18,133)(19,104)(20,135)(21,46)(22,77)(23,48)(24,79)(25,50)(26,71)(27,42)(28,73)(29,44)(30,75)(31,91)(32,83)(33,93)(34,85)(35,95)(36,87)(37,97)(38,89)(39,99)(40,81)(41,56)(43,58)(45,60)(47,52)(49,54)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(62,92)(63,84)(64,94)(65,86)(66,96)(67,88)(68,98)(69,90)(70,100)(101,156)(103,158)(105,160)(107,152)(109,154)(112,127)(114,129)(116,121)(118,123)(120,125)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,14,151,154)(12,153,152,13)(15,20,155,160)(16,159,156,19)(17,18,157,158)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,68,61,38)(32,37,62,67)(33,66,63,36)(34,35,64,65)(39,70,69,40)(41,78,71,48)(42,47,72,77)(43,76,73,46)(44,45,74,75)(49,80,79,50)(81,99,100,90)(82,89,91,98)(83,97,92,88)(84,87,93,96)(85,95,94,86)(101,104,131,134)(102,133,132,103)(105,110,135,140)(106,139,136,109)(107,108,137,138)(111,114,141,144)(112,143,142,113)(115,120,145,150)(116,149,146,119)(117,118,147,148) );

G=PermutationGroup([(1,95,30,153,123,86,60,13),(2,96,21,154,124,87,51,14),(3,97,22,155,125,88,52,15),(4,98,23,156,126,89,53,16),(5,99,24,157,127,90,54,17),(6,100,25,158,128,81,55,18),(7,91,26,159,129,82,56,19),(8,92,27,160,130,83,57,20),(9,93,28,151,121,84,58,11),(10,94,29,152,122,85,59,12),(31,144,104,41,61,114,134,71),(32,145,105,42,62,115,135,72),(33,146,106,43,63,116,136,73),(34,147,107,44,64,117,137,74),(35,148,108,45,65,118,138,75),(36,149,109,46,66,119,139,76),(37,150,110,47,67,120,140,77),(38,141,101,48,68,111,131,78),(39,142,102,49,69,112,132,79),(40,143,103,50,70,113,133,80)], [(1,148),(2,119),(3,150),(4,111),(5,142),(6,113),(7,144),(8,115),(9,146),(10,117),(11,106),(12,137),(13,108),(14,139),(15,110),(16,131),(17,102),(18,133),(19,104),(20,135),(21,46),(22,77),(23,48),(24,79),(25,50),(26,71),(27,42),(28,73),(29,44),(30,75),(31,91),(32,83),(33,93),(34,85),(35,95),(36,87),(37,97),(38,89),(39,99),(40,81),(41,56),(43,58),(45,60),(47,52),(49,54),(51,76),(53,78),(55,80),(57,72),(59,74),(61,82),(62,92),(63,84),(64,94),(65,86),(66,96),(67,88),(68,98),(69,90),(70,100),(101,156),(103,158),(105,160),(107,152),(109,154),(112,127),(114,129),(116,121),(118,123),(120,125),(122,147),(124,149),(126,141),(128,143),(130,145),(132,157),(134,159),(136,151),(138,153),(140,155)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,14,151,154),(12,153,152,13),(15,20,155,160),(16,159,156,19),(17,18,157,158),(21,58,51,28),(22,27,52,57),(23,56,53,26),(24,25,54,55),(29,60,59,30),(31,68,61,38),(32,37,62,67),(33,66,63,36),(34,35,64,65),(39,70,69,40),(41,78,71,48),(42,47,72,77),(43,76,73,46),(44,45,74,75),(49,80,79,50),(81,99,100,90),(82,89,91,98),(83,97,92,88),(84,87,93,96),(85,95,94,86),(101,104,131,134),(102,133,132,103),(105,110,135,140),(106,139,136,109),(107,108,137,138),(111,114,141,144),(112,143,142,113),(115,120,145,150),(116,149,146,119),(117,118,147,148)])

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222224444444444558888810101010101010102020202020202020202040···40
size11244101022441010202020202222420202244888822224488884···4

50 irreducible representations

dim1111111112222222224444
type++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10Q8○D8D4×D5D4×D5D8.10D10
kernelD8.10D10D20.3C4C2×Dic20D83D5SD16⋊D5D5×Q16D4.9D10C5×C4○D8D4.10D10Dic10D20C5⋊D4C4○D8C2×C8D8SD16Q16C4○D4C5C4C22C1
# reps1112422121122224242228

Matrix representation of D8.10D10 in GL4(𝔽41) generated by

00338
00330
036240
536024
,
5293128
12361018
6183412
2424277
,
39162315
2525638
773225
114527
,
39132115
2521838
7351425
1332527
G:=sub<GL(4,GF(41))| [0,0,0,5,0,0,36,36,33,33,24,0,8,0,0,24],[5,12,6,24,29,36,18,24,31,10,34,27,28,18,12,7],[39,25,7,1,16,25,7,14,23,6,32,5,15,38,25,27],[39,25,7,1,13,2,35,33,21,18,14,25,15,38,25,27] >;

D8.10D10 in GAP, Magma, Sage, TeX

D_8._{10}D_{10}
% in TeX

G:=Group("D8.10D10");
// GroupNames label

G:=SmallGroup(320,1443);
// by ID

G=gap.SmallGroup(320,1443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,184,570,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations

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