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G = D206Q8order 320 = 26·5

4th semidirect product of D20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D206Q8, C20.18D8, C42.81D10, C4⋊Q85D5, C4.12(Q8×D5), C4⋊C4.83D10, C55(D4⋊Q8), C203C835C2, C10.60(C2×D8), C20.39(C2×Q8), C4.16(D4⋊D5), (C4×D20).19C2, (C2×C20).157D4, C20.82(C4○D4), C10.D843C2, D206C4.15C2, (C4×C20).134C22, (C2×C20).405C23, C4.35(Q82D5), C10.76(C22⋊Q8), C2.13(D103Q8), (C2×D20).257C22, C10.97(C8.C22), C4⋊Dic5.348C22, C2.18(C20.C23), (C5×C4⋊Q8)⋊5C2, C2.15(C2×D4⋊D5), (C2×C10).536(C2×D4), (C2×C4).189(C5⋊D4), (C5×C4⋊C4).130C22, (C2×C4).502(C22×D5), C22.208(C2×C5⋊D4), (C2×C52C8).138C22, SmallGroup(320,714)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D206Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D206Q8
C5C10C2×C20 — D206Q8
C1C22C42C4⋊Q8

Generators and relations for D206Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 438 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C52C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C22×D5, D4⋊Q8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C203C8, C10.D8 [×2], D206C4 [×2], C4×D20, C5×C4⋊Q8, D206Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, D8 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×D8, C8.C22, C5⋊D4 [×2], C22×D5, D4⋊Q8, D4⋊D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, C2×D4⋊D5, C20.C23, D103Q8, D206Q8

Smallest permutation representation of D206Q8
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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 124)(22 123)(23 122)(24 121)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 160)(48 159)(49 158)(50 157)(51 156)(52 155)(53 154)(54 153)(55 152)(56 151)(57 150)(58 149)(59 148)(60 147)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)
(1 122 111 39)(2 133 112 30)(3 124 113 21)(4 135 114 32)(5 126 115 23)(6 137 116 34)(7 128 117 25)(8 139 118 36)(9 130 119 27)(10 121 120 38)(11 132 101 29)(12 123 102 40)(13 134 103 31)(14 125 104 22)(15 136 105 33)(16 127 106 24)(17 138 107 35)(18 129 108 26)(19 140 109 37)(20 131 110 28)(41 99 151 74)(42 90 152 65)(43 81 153 76)(44 92 154 67)(45 83 155 78)(46 94 156 69)(47 85 157 80)(48 96 158 71)(49 87 159 62)(50 98 160 73)(51 89 141 64)(52 100 142 75)(53 91 143 66)(54 82 144 77)(55 93 145 68)(56 84 146 79)(57 95 147 70)(58 86 148 61)(59 97 149 72)(60 88 150 63)
(1 49 111 159)(2 50 112 160)(3 51 113 141)(4 52 114 142)(5 53 115 143)(6 54 116 144)(7 55 117 145)(8 56 118 146)(9 57 119 147)(10 58 120 148)(11 59 101 149)(12 60 102 150)(13 41 103 151)(14 42 104 152)(15 43 105 153)(16 44 106 154)(17 45 107 155)(18 46 108 156)(19 47 109 157)(20 48 110 158)(21 89 124 64)(22 90 125 65)(23 91 126 66)(24 92 127 67)(25 93 128 68)(26 94 129 69)(27 95 130 70)(28 96 131 71)(29 97 132 72)(30 98 133 73)(31 99 134 74)(32 100 135 75)(33 81 136 76)(34 82 137 77)(35 83 138 78)(36 84 139 79)(37 85 140 80)(38 86 121 61)(39 87 122 62)(40 88 123 63)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12222244444444455888810···1020···2020···20
size111120202222488202022202020202···24···48···8

47 irreducible representations

dim1111112222222244444
type++++++-+++++-+-+
imageC1C2C2C2C2C2Q8D4D5D8C4○D4D10D10C5⋊D4C8.C22D4⋊D5Q8×D5Q82D5C20.C23
kernelD206Q8C203C8C10.D8D206C4C4×D20C5×C4⋊Q8D20C2×C20C4⋊Q8C20C20C42C4⋊C4C2×C4C10C4C4C4C2
# reps1122112224224814224

Matrix representation of D206Q8 in GL6(𝔽41)

010000
4060000
0013700
00214000
0000400
0000040
,
010000
100000
0013700
0004000
0000123
0000040
,
100000
010000
0017700
00352400
000090
0000132
,
100000
010000
001000
000100
00003239
000009

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,21,0,0,0,0,37,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,37,40,0,0,0,0,0,0,1,0,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,35,0,0,0,0,7,24,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,39,9] >;

D206Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_6Q_8
% in TeX

G:=Group("D20:6Q8");
// GroupNames label

G:=SmallGroup(320,714);
// by ID

G=gap.SmallGroup(320,714);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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