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G = D20.4Q8order 320 = 26·5

2nd non-split extension by D20 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.4Q8, C42.69D10, C57(D4.Q8), C4.10(Q8×D5), C4⋊C4.75D10, C203C830C2, C20.34(C2×Q8), C42.C21D5, (C4×D20).16C2, (C2×C20).275D4, C20.70(C4○D4), C10.D842C2, C20.Q841C2, D206C4.12C2, C10.109(C4○D8), (C4×C20).114C22, (C2×C20).384C23, C4.33(Q82D5), C10.74(C22⋊Q8), C2.21(D4⋊D10), C10.122(C8⋊C22), C2.11(D103Q8), (C2×D20).253C22, C4⋊Dic5.343C22, C2.28(D4.8D10), (C5×C42.C2)⋊1C2, (C2×C10).515(C2×D4), (C2×C4).66(C5⋊D4), (C5×C4⋊C4).122C22, (C2×C4).482(C22×D5), C22.188(C2×C5⋊D4), (C2×C52C8).126C22, SmallGroup(320,693)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.4Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D20.4Q8
C5C10C2×C20 — D20.4Q8
C1C22C42C42.C2

Generators and relations for D20.4Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=a10c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 422 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, Dic5, C20 [×2], C20 [×4], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C22×D5, D4.Q8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×D20, C203C8, C10.D8, C20.Q8, D206C4 [×2], C4×D20, C5×C42.C2, D20.4Q8
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, C5⋊D4 [×2], C22×D5, D4.Q8, Q8×D5, Q82D5, C2×C5⋊D4, D103Q8, D4⋊D10, D4.8D10, D20.4Q8

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim11111112222222244444
type+++++++-+++++-++
imageC1C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C5⋊D4C8⋊C22Q8×D5Q82D5D4⋊D10D4.8D10
kernelD20.4Q8C203C8C10.D8C20.Q8D206C4C4×D20C5×C42.C2D20C2×C20C42.C2C20C42C4⋊C4C10C2×C4C10C4C4C2C2
# reps11112112222244812244

Matrix representation of D20.4Q8 in GL6(𝔽41)

010000
4060000
001000
000100
0000139
0000140
,
010000
100000
0040000
0004000
0000402
000001
,
4000000
0400000
000100
0040000
00002417
00001217
,
4000000
0400000
0063900
00393500
0000320
0000032

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,12,0,0,0,0,17,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,39,0,0,0,0,39,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;

D20.4Q8 in GAP, Magma, Sage, TeX

D_{20}._4Q_8
% in TeX

G:=Group("D20.4Q8");
// GroupNames label

G:=SmallGroup(320,693);
// by ID

G=gap.SmallGroup(320,693);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10*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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