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G = D10.8Q16order 320 = 26·5

2nd non-split extension by D10 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.8Q16, C2.D84D5, C4⋊C4.49D10, (C2×C8).28D10, C2.14(D5×Q16), C10.24(C2×Q16), C20.39(C4○D4), C4.81(C4○D20), C10.D822C2, C10.Q1619C2, D102Q8.8C2, (C2×Dic5).57D4, C22.230(D4×D5), D101C8.10C2, C2.22(D8⋊D5), C20.44D426C2, C10.41(C8⋊C22), (C2×C40).242C22, (C2×C20).300C23, C4.29(Q82D5), (C22×D5).123D4, C53(C23.48D4), C4⋊Dic5.125C22, (C2×Dic10).93C22, C2.16(D10.13D4), C10.46(C22.D4), (D5×C4⋊C4).9C2, (C5×C2.D8)⋊12C2, (C2×C4×D5).43C22, (C2×C10).305(C2×D4), (C5×C4⋊C4).93C22, (C2×C52C8).70C22, (C2×C4).403(C22×D5), SmallGroup(320,511)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.8Q16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.8Q16
C5C10C2×C20 — D10.8Q16
C1C22C2×C4C2.D8

Generators and relations for D10.8Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=a5c4, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 430 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×9], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C22×C4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4 [×2], C2.D8, C2.D8, C2×C4⋊C4, C22⋊Q8, C52C8, C40, Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.48D4, C2×C52C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, C10.D8, C10.Q16, C20.44D4, D101C8, C5×C2.D8, D5×C4⋊C4, D102Q8, D10.8Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×Q16, C8⋊C22, C22×D5, C23.48D4, C4○D20, D4×D5, Q82D5, D10.13D4, D8⋊D5, D5×Q16, D10.8Q16

Smallest permutation representation of D10.8Q16
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 134)(22 133)(23 132)(24 131)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 90)(60 89)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(91 123)(92 122)(93 121)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(110 120)(141 154)(142 153)(143 152)(144 151)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)
(1 85 45 68 11 60 37 73)(2 86 46 69 12 51 38 74)(3 87 47 70 13 52 39 75)(4 88 48 61 14 53 40 76)(5 89 49 62 15 54 31 77)(6 90 50 63 16 55 32 78)(7 81 41 64 17 56 33 79)(8 82 42 65 18 57 34 80)(9 83 43 66 19 58 35 71)(10 84 44 67 20 59 36 72)(21 113 153 100 140 108 148 125)(22 114 154 91 131 109 149 126)(23 115 155 92 132 110 150 127)(24 116 156 93 133 101 141 128)(25 117 157 94 134 102 142 129)(26 118 158 95 135 103 143 130)(27 119 159 96 136 104 144 121)(28 120 160 97 137 105 145 122)(29 111 151 98 138 106 146 123)(30 112 152 99 139 107 147 124)
(1 103 16 113)(2 104 17 114)(3 105 18 115)(4 106 19 116)(5 107 20 117)(6 108 11 118)(7 109 12 119)(8 110 13 120)(9 101 14 111)(10 102 15 112)(21 85 135 55)(22 86 136 56)(23 87 137 57)(24 88 138 58)(25 89 139 59)(26 90 140 60)(27 81 131 51)(28 82 132 52)(29 83 133 53)(30 84 134 54)(31 124 44 94)(32 125 45 95)(33 126 46 96)(34 127 47 97)(35 128 48 98)(36 129 49 99)(37 130 50 100)(38 121 41 91)(39 122 42 92)(40 123 43 93)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111110102244820202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++-+++++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4Q16D10D10C4○D20C8⋊C22Q82D5D4×D5D8⋊D5D5×Q16
kernelD10.8Q16C10.D8C10.Q16C20.44D4D101C8C5×C2.D8D5×C4⋊C4D102Q8C2×Dic5C22×D5C2.D8C20D10C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111111124442812244

Matrix representation of D10.8Q16 in GL4(𝔽41) generated by

343400
7100
0010
0001
,
343400
1700
00400
00040
,
174000
12400
0007
003524
,
9000
0900
00172
001924
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,1,0,0,0,0,1],[34,1,0,0,34,7,0,0,0,0,40,0,0,0,0,40],[17,1,0,0,40,24,0,0,0,0,0,35,0,0,7,24],[9,0,0,0,0,9,0,0,0,0,17,19,0,0,2,24] >;

D10.8Q16 in GAP, Magma, Sage, TeX

D_{10}._8Q_{16}
% in TeX

G:=Group("D10.8Q16");
// GroupNames label

G:=SmallGroup(320,511);
// by ID

G=gap.SmallGroup(320,511);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,268,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5*c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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