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G = D20.2Q8order 320 = 26·5

2nd non-split extension by D20 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.2Q8, C4.8(Q8×D5), C2.D89D5, C55(D4.Q8), (C2×C8).32D10, C20.24(C2×Q8), C4⋊C4.174D10, C4.Dic107C2, D206C4.9C2, D208C4.9C2, C4.85(C4○D20), C10.77(C4○D8), C20.Q823C2, C20.8Q826C2, (C2×Dic5).60D4, C22.236(D4×D5), D205C4.11C2, C20.173(C4○D4), C2.25(D8⋊D5), C10.45(C8⋊C22), (C2×C40).246C22, (C2×C20).307C23, (C2×D20).91C22, C10.42(C22⋊Q8), C2.15(Q8.D10), C2.19(D10⋊Q8), C4⋊Dic5.129C22, (C4×Dic5).45C22, (C5×C2.D8)⋊16C2, (C2×C10).312(C2×D4), (C5×C4⋊C4).100C22, (C2×C52C8).76C22, (C2×C4).410(C22×D5), SmallGroup(320,518)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.2Q8
C1C5C10C2×C10C2×C20C2×D20D208C4 — D20.2Q8
C5C10C2×C20 — D20.2Q8
C1C22C2×C4C2.D8

Generators and relations for D20.2Q8
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=a10c3 >

Subgroups: 454 in 102 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 [×7], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×D20, C20.Q8, D206C4, C20.8Q8, D205C4, C5×C2.D8, C4.Dic10, D208C4, D20.2Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, C22×D5, D4.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D8⋊D5, Q8.D10, D20.2Q8

Smallest permutation representation of D20.2Q8
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 120)(40 119)(61 82)(62 81)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(121 152)(122 151)(123 150)(124 149)(125 148)(126 147)(127 146)(128 145)(129 144)(130 143)(131 142)(132 141)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)
(1 130 49 159 11 140 59 149)(2 131 50 160 12 121 60 150)(3 132 51 141 13 122 41 151)(4 133 52 142 14 123 42 152)(5 134 53 143 15 124 43 153)(6 135 54 144 16 125 44 154)(7 136 55 145 17 126 45 155)(8 137 56 146 18 127 46 156)(9 138 57 147 19 128 47 157)(10 139 58 148 20 129 48 158)(21 66 113 87 31 76 103 97)(22 67 114 88 32 77 104 98)(23 68 115 89 33 78 105 99)(24 69 116 90 34 79 106 100)(25 70 117 91 35 80 107 81)(26 71 118 92 36 61 108 82)(27 72 119 93 37 62 109 83)(28 73 120 94 38 63 110 84)(29 74 101 95 39 64 111 85)(30 75 102 96 40 65 112 86)
(1 101 44 34)(2 112 45 25)(3 103 46 36)(4 114 47 27)(5 105 48 38)(6 116 49 29)(7 107 50 40)(8 118 51 31)(9 109 52 22)(10 120 53 33)(11 111 54 24)(12 102 55 35)(13 113 56 26)(14 104 57 37)(15 115 58 28)(16 106 59 39)(17 117 60 30)(18 108 41 21)(19 119 42 32)(20 110 43 23)(61 151 97 127)(62 142 98 138)(63 153 99 129)(64 144 100 140)(65 155 81 131)(66 146 82 122)(67 157 83 133)(68 148 84 124)(69 159 85 135)(70 150 86 126)(71 141 87 137)(72 152 88 128)(73 143 89 139)(74 154 90 130)(75 145 91 121)(76 156 92 132)(77 147 93 123)(78 158 94 134)(79 149 95 125)(80 160 96 136)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120202244810102040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++-+++++-++
imageC1C2C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C4○D20C8⋊C22Q8×D5D4×D5D8⋊D5Q8.D10
kernelD20.2Q8C20.Q8D206C4C20.8Q8D205C4C5×C2.D8C4.Dic10D208C4D20C2×Dic5C2.D8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111112222424812244

Matrix representation of D20.2Q8 in GL6(𝔽41)

100000
010000
000100
0040600
0000121
00003740
,
4000000
0400000
000100
001000
00004020
000001
,
010000
4000000
001000
000100
000006
00003417
,
3200000
090000
001000
000100
00003216
000009

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,20,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,0,0,0,0,6,17],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,16,9] >;

D20.2Q8 in GAP, Magma, Sage, TeX

D_{20}._2Q_8
% in TeX

G:=Group("D20.2Q8");
// GroupNames label

G:=SmallGroup(320,518);
// by ID

G=gap.SmallGroup(320,518);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,100,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^10*c^3>;
// generators/relations

׿
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