Copied to
clipboard

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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D42Q8, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C406C4, D205C4, C5×C4⋊C8, C202Q8, C4×D20, D203Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, D20, C22×D5, D42Q8, C40⋊C2, 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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(41 125)(42 124)(43 123)(44 122)(45 121)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 83)(62 82)(63 81)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(101 158)(102 157)(103 156)(104 155)(105 154)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 144)(116 143)(117 142)(118 141)(119 160)(120 159)
(1 77 37 93)(2 76 38 92)(3 75 39 91)(4 74 40 90)(5 73 21 89)(6 72 22 88)(7 71 23 87)(8 70 24 86)(9 69 25 85)(10 68 26 84)(11 67 27 83)(12 66 28 82)(13 65 29 81)(14 64 30 100)(15 63 31 99)(16 62 32 98)(17 61 33 97)(18 80 34 96)(19 79 35 95)(20 78 36 94)(41 109 130 142)(42 108 131 141)(43 107 132 160)(44 106 133 159)(45 105 134 158)(46 104 135 157)(47 103 136 156)(48 102 137 155)(49 101 138 154)(50 120 139 153)(51 119 140 152)(52 118 121 151)(53 117 122 150)(54 116 123 149)(55 115 124 148)(56 114 125 147)(57 113 126 146)(58 112 127 145)(59 111 128 144)(60 110 129 143)
(1 49 37 138)(2 50 38 139)(3 51 39 140)(4 52 40 121)(5 53 21 122)(6 54 22 123)(7 55 23 124)(8 56 24 125)(9 57 25 126)(10 58 26 127)(11 59 27 128)(12 60 28 129)(13 41 29 130)(14 42 30 131)(15 43 31 132)(16 44 32 133)(17 45 33 134)(18 46 34 135)(19 47 35 136)(20 48 36 137)(61 158 97 105)(62 159 98 106)(63 160 99 107)(64 141 100 108)(65 142 81 109)(66 143 82 110)(67 144 83 111)(68 145 84 112)(69 146 85 113)(70 147 86 114)(71 148 87 115)(72 149 88 116)(73 150 89 117)(74 151 90 118)(75 152 91 119)(76 153 92 120)(77 154 93 101)(78 155 94 102)(79 156 95 103)(80 157 96 104)

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

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

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

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

׿
×
𝔽