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G = D203Q8order 320 = 26·5

1st semidirect product of D20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D203Q8, C20.13SD16, C42.33D10, C4⋊C89D5, C4.43(Q8×D5), C52(D42Q8), C406C416C2, C202Q812C2, (C4×D20).11C2, (C2×C8).129D10, (C2×C20).121D4, (C2×C4).132D20, C20.102(C2×Q8), D205C4.5C2, C4.13(C40⋊C2), (C4×C20).68C22, C10.11(C2×SD16), C20.286(C4○D4), C2.16(C8⋊D10), C10.13(C8⋊C22), (C2×C40).139C22, (C2×C20).752C23, C22.115(C2×D20), C10.30(C22⋊Q8), C4⋊Dic5.18C22, C4.110(D42D5), C2.11(D102Q8), (C2×D20).201C22, (C5×C4⋊C8)⋊11C2, C2.14(C2×C40⋊C2), (C2×C10).135(C2×D4), (C2×C4).697(C22×D5), SmallGroup(320,469)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D203Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D203Q8
C5C10C2×C20 — D203Q8
C1C22C42C4⋊C8

Generators and relations for D203Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 518 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 [×4], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20, D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, D42Q8, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20, C406C4 [×2], D205C4 [×2], C5×C4⋊C8, C202Q8, C4×D20, D203Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8⋊C22, D20 [×2], C22×D5, D42Q8, C40⋊C2 [×2], C2×D20, D42D5, Q8×D5, D102Q8, C2×C40⋊C2, C8⋊D10, D203Q8

Smallest permutation representation of D203Q8
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)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 160)(11 159)(12 158)(13 157)(14 156)(15 155)(16 154)(17 153)(18 152)(19 151)(20 150)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 121)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)
(1 103 160 93)(2 102 141 92)(3 101 142 91)(4 120 143 90)(5 119 144 89)(6 118 145 88)(7 117 146 87)(8 116 147 86)(9 115 148 85)(10 114 149 84)(11 113 150 83)(12 112 151 82)(13 111 152 81)(14 110 153 100)(15 109 154 99)(16 108 155 98)(17 107 156 97)(18 106 157 96)(19 105 158 95)(20 104 159 94)(21 133 69 59)(22 132 70 58)(23 131 71 57)(24 130 72 56)(25 129 73 55)(26 128 74 54)(27 127 75 53)(28 126 76 52)(29 125 77 51)(30 124 78 50)(31 123 79 49)(32 122 80 48)(33 121 61 47)(34 140 62 46)(35 139 63 45)(36 138 64 44)(37 137 65 43)(38 136 66 42)(39 135 67 41)(40 134 68 60)
(1 66 160 38)(2 67 141 39)(3 68 142 40)(4 69 143 21)(5 70 144 22)(6 71 145 23)(7 72 146 24)(8 73 147 25)(9 74 148 26)(10 75 149 27)(11 76 150 28)(12 77 151 29)(13 78 152 30)(14 79 153 31)(15 80 154 32)(16 61 155 33)(17 62 156 34)(18 63 157 35)(19 64 158 36)(20 65 159 37)(41 102 135 92)(42 103 136 93)(43 104 137 94)(44 105 138 95)(45 106 139 96)(46 107 140 97)(47 108 121 98)(48 109 122 99)(49 110 123 100)(50 111 124 81)(51 112 125 82)(52 113 126 83)(53 114 127 84)(54 115 128 85)(55 116 129 86)(56 117 130 87)(57 118 131 88)(58 119 132 89)(59 120 133 90)(60 101 134 91)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222244444444455888810···1020···2020···2040···40
size1111202022224202040402244442···22···24···44···4

59 irreducible representations

dim1111112222222224444
type++++++-++++++--+
imageC1C2C2C2C2C2Q8D4D5SD16C4○D4D10D10D20C40⋊C2C8⋊C22D42D5Q8×D5C8⋊D10
kernelD203Q8C406C4D205C4C5×C4⋊C8C202Q8C4×D20D20C2×C20C4⋊C8C20C20C42C2×C8C2×C4C4C10C4C4C2
# reps12211122242248161224

Matrix representation of D203Q8 in GL4(𝔽41) generated by

283900
21600
00400
00040
,
283900
21300
0010
001140
,
371000
27400
00302
002111
,
392800
13200
00320
00249
G:=sub<GL(4,GF(41))| [28,2,0,0,39,16,0,0,0,0,40,0,0,0,0,40],[28,2,0,0,39,13,0,0,0,0,1,11,0,0,0,40],[37,27,0,0,10,4,0,0,0,0,30,21,0,0,2,11],[39,13,0,0,28,2,0,0,0,0,32,24,0,0,0,9] >;

D203Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_3Q_8
% in TeX

G:=Group("D20:3Q8");
// GroupNames label

G:=SmallGroup(320,469);
// by ID

G=gap.SmallGroup(320,469);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,142,1123,136,12550]);
// Polycyclic

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

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