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G = D205Q8order 320 = 26·5

3rd semidirect product of D20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D205Q8, C20.18SD16, C42.79D10, C4⋊Q82D5, C4.11(Q8×D5), C4⋊C4.82D10, C55(D42Q8), C203C834C2, C20.38(C2×Q8), (C4×D20).18C2, (C2×C20).155D4, C4.10(Q8⋊D5), C20.81(C4○D4), C20.Q842C2, C10.76(C2×SD16), D206C4.14C2, C10.98(C8⋊C22), (C4×C20).131C22, (C2×C20).402C23, C4.34(Q82D5), C10.75(C22⋊Q8), C2.12(D103Q8), (C2×D20).256C22, C4⋊Dic5.347C22, C2.19(D4.D10), (C5×C4⋊Q8)⋊2C2, C2.14(C2×Q8⋊D5), (C2×C10).533(C2×D4), (C2×C4).188(C5⋊D4), (C5×C4⋊C4).129C22, (C2×C4).499(C22×D5), C22.205(C2×C5⋊D4), (C2×C52C8).136C22, SmallGroup(320,711)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D205Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D205Q8
C5C10C2×C20 — D205Q8
C1C22C42C4⋊Q8

Generators and relations for D205Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a10b, 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, C4.Q8 [×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, D42Q8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C203C8, C20.Q8 [×2], D206C4 [×2], C4×D20, C5×C4⋊Q8, D205Q8
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, C5⋊D4 [×2], C22×D5, D42Q8, Q8⋊D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, D4.D10, C2×Q8⋊D5, D103Q8, D205Q8

Smallest permutation representation of D205Q8
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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 141)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)
(1 34 102 134)(2 25 103 125)(3 36 104 136)(4 27 105 127)(5 38 106 138)(6 29 107 129)(7 40 108 140)(8 31 109 131)(9 22 110 122)(10 33 111 133)(11 24 112 124)(12 35 113 135)(13 26 114 126)(14 37 115 137)(15 28 116 128)(16 39 117 139)(17 30 118 130)(18 21 119 121)(19 32 120 132)(20 23 101 123)(41 155 87 63)(42 146 88 74)(43 157 89 65)(44 148 90 76)(45 159 91 67)(46 150 92 78)(47 141 93 69)(48 152 94 80)(49 143 95 71)(50 154 96 62)(51 145 97 73)(52 156 98 64)(53 147 99 75)(54 158 100 66)(55 149 81 77)(56 160 82 68)(57 151 83 79)(58 142 84 70)(59 153 85 61)(60 144 86 72)
(1 87 102 41)(2 88 103 42)(3 89 104 43)(4 90 105 44)(5 91 106 45)(6 92 107 46)(7 93 108 47)(8 94 109 48)(9 95 110 49)(10 96 111 50)(11 97 112 51)(12 98 113 52)(13 99 114 53)(14 100 115 54)(15 81 116 55)(16 82 117 56)(17 83 118 57)(18 84 119 58)(19 85 120 59)(20 86 101 60)(21 142 121 70)(22 143 122 71)(23 144 123 72)(24 145 124 73)(25 146 125 74)(26 147 126 75)(27 148 127 76)(28 149 128 77)(29 150 129 78)(30 151 130 79)(31 152 131 80)(32 153 132 61)(33 154 133 62)(34 155 134 63)(35 156 135 64)(36 157 136 65)(37 158 137 66)(38 159 138 67)(39 160 139 68)(40 141 140 69)

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim1111112222222244444
type++++++-++++++-+
imageC1C2C2C2C2C2Q8D4D5SD16C4○D4D10D10C5⋊D4C8⋊C22Q8⋊D5Q8×D5Q82D5D4.D10
kernelD205Q8C203C8C20.Q8D206C4C4×D20C5×C4⋊Q8D20C2×C20C4⋊Q8C20C20C42C4⋊C4C2×C4C10C4C4C4C2
# reps1122112224224814224

Matrix representation of D205Q8 in GL6(𝔽41)

4000000
0400000
0035100
0054000
0000404
0000201
,
100000
32400000
00404000
000100
000010
00002140
,
250000
40390000
001000
000100
00001119
00001330
,
3200000
4090000
0040000
0004000
0000137
00002140

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,20,0,0,0,0,4,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[2,40,0,0,0,0,5,39,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,13,0,0,0,0,19,30],[32,40,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,37,40] >;

D205Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_5Q_8
% in TeX

G:=Group("D20:5Q8");
// GroupNames label

G:=SmallGroup(320,711);
// by ID

G=gap.SmallGroup(320,711);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,268,1123,297,136,12550]);
// 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,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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