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G = D10⋊D8order 320 = 26·5

1st semidirect product of D10 and D8 acting via D8/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D101D8, C52C819D4, C51(C87D4), C4⋊C4.9D10, C2.10(D5×D8), D4⋊C42D5, C202D41C2, C4⋊D202C2, C10.24(C2×D8), C4.158(D4×D5), (C2×D4).24D10, C20.7(C4○D4), C10.D86C2, (C2×C8).202D10, C20.108(C2×D4), D205C419C2, C4.24(C4○D20), C10.40(C4○D8), (C22×D5).78D4, C22.173(D4×D5), C10.15(C4⋊D4), (C2×C40).184C22, (C2×C20).215C23, (C2×Dic5).134D4, (C2×D20).53C22, (D4×C10).36C22, C4⋊Dic5.70C22, C2.18(D10⋊D4), C2.10(SD163D5), (D5×C2×C8)⋊18C2, (C2×D4⋊D5)⋊3C2, (C5×D4⋊C4)⋊22C2, (C2×C10).228(C2×D4), (C5×C4⋊C4).17C22, (C2×C4×D5).294C22, (C2×C4).322(C22×D5), (C2×C52C8).220C22, SmallGroup(320,402)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10⋊D8
C1C5C10C20C2×C20C2×C4×D5D5×C2×C8 — D10⋊D8
C5C10C2×C20 — D10⋊D8
C1C22C2×C4D4⋊C4

Generators and relations for D10⋊D8
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >

Subgroups: 662 in 134 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C8 [×3], C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5 [×3], C10 [×3], C10, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8 [×3], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], D4⋊C4, D4⋊C4, C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, C52C8 [×2], C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×D5, C22×C10, C87D4, C8×D5 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, D4×C10, C10.D8, D205C4, C5×D4⋊C4, C4⋊D20, D5×C2×C8, C2×D4⋊D5, C202D4, D10⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, C22×D5, C87D4, C4○D20, D4×D5 [×2], D10⋊D4, D5×D8, SD163D5, D10⋊D8

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444558888888810···1010101010202020202020202040···40
size11118101040228101040222222101010102···28888444488884···4

50 irreducible representations

dim11111111222222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D8D10D10D10C4○D8C4○D20D4×D5D4×D5D5×D8SD163D5
kernelD10⋊D8C10.D8D205C4C5×D4⋊C4C4⋊D20D5×C2×C8C2×D4⋊D5C202D4C52C8C2×Dic5C22×D5D4⋊C4C20D10C4⋊C4C2×C8C2×D4C10C4C4C22C2C2
# reps11111111211224222482244

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

1000
0100
00134
00734
,
40000
04000
00309
001411
,
291200
292900
003838
00173
,
0100
1000
00347
00407
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,7,0,0,34,34],[40,0,0,0,0,40,0,0,0,0,30,14,0,0,9,11],[29,29,0,0,12,29,0,0,0,0,38,17,0,0,38,3],[0,1,0,0,1,0,0,0,0,0,34,40,0,0,7,7] >;

D10⋊D8 in GAP, Magma, Sage, TeX

D_{10}\rtimes D_8
% in TeX

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

G:=SmallGroup(320,402);
// by ID

G=gap.SmallGroup(320,402);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,297,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

׿
×
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