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G = Q82D20order 320 = 26·5

1st semidirect product of Q8 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82D20, D107SD16, Dic104D4, (C5×Q8)⋊1D4, C4.5(C2×D20), C4.92(D4×D5), C4⋊C4.23D10, C52(Q8⋊D4), Q8⋊C411D5, C20.120(C2×D4), C4⋊D20.2C2, (C2×C8).123D10, D101C813C2, C2.17(D5×SD16), C10.23C22≀C2, C10.Q1611C2, (C2×Q8).107D10, (C2×Dic5).38D4, C10.31(C2×SD16), C22.196(D4×D5), (C2×C40).134C22, (C2×C20).246C23, (C22×D5).114D4, (C2×D20).66C22, (Q8×C10).29C22, C2.26(C22⋊D20), C2.14(Q16⋊D5), C10.60(C8.C22), (C2×Dic10).74C22, (C2×Q8×D5)⋊1C2, (C2×Q8⋊D5)⋊2C2, (C2×C40⋊C2)⋊17C2, (C2×C4×D5).24C22, (C5×Q8⋊C4)⋊11C2, (C2×C10).259(C2×D4), (C5×C4⋊C4).47C22, (C2×C52C8).38C22, (C2×C4).353(C22×D5), SmallGroup(320,433)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Q82D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×Q8×D5 — Q82D20
C5C10C2×C20 — Q82D20
C1C22C2×C4Q8⋊C4

Generators and relations for Q82D20
 G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=ab, dcd=c-1 >

Subgroups: 750 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], Q8 [×2], Q8 [×8], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×6], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4⋊D4, C2×SD16 [×2], C22×Q8, C52C8, C40, Dic10 [×2], Dic10 [×5], C4×D5 [×6], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, Q8⋊D4, C40⋊C2 [×2], C2×C52C8, D10⋊C4, Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×D5 [×4], Q8×C10, C10.Q16, D101C8, C5×Q8⋊C4, C4⋊D20, C2×C40⋊C2, C2×Q8⋊D5, C2×Q8×D5, Q82D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, D20 [×2], C22×D5, Q8⋊D4, C2×D20, D4×D5 [×2], C22⋊D20, D5×SD16, Q16⋊D5, Q82D20

Smallest permutation representation of Q82D20
On 160 points
Generators in S160
(1 132 69 90)(2 91 70 133)(3 134 71 92)(4 93 72 135)(5 136 73 94)(6 95 74 137)(7 138 75 96)(8 97 76 139)(9 140 77 98)(10 99 78 121)(11 122 79 100)(12 81 80 123)(13 124 61 82)(14 83 62 125)(15 126 63 84)(16 85 64 127)(17 128 65 86)(18 87 66 129)(19 130 67 88)(20 89 68 131)(21 154 105 59)(22 60 106 155)(23 156 107 41)(24 42 108 157)(25 158 109 43)(26 44 110 159)(27 160 111 45)(28 46 112 141)(29 142 113 47)(30 48 114 143)(31 144 115 49)(32 50 116 145)(33 146 117 51)(34 52 118 147)(35 148 119 53)(36 54 120 149)(37 150 101 55)(38 56 102 151)(39 152 103 57)(40 58 104 153)
(1 115 69 31)(2 145 70 50)(3 117 71 33)(4 147 72 52)(5 119 73 35)(6 149 74 54)(7 101 75 37)(8 151 76 56)(9 103 77 39)(10 153 78 58)(11 105 79 21)(12 155 80 60)(13 107 61 23)(14 157 62 42)(15 109 63 25)(16 159 64 44)(17 111 65 27)(18 141 66 46)(19 113 67 29)(20 143 68 48)(22 81 106 123)(24 83 108 125)(26 85 110 127)(28 87 112 129)(30 89 114 131)(32 91 116 133)(34 93 118 135)(36 95 120 137)(38 97 102 139)(40 99 104 121)(41 124 156 82)(43 126 158 84)(45 128 160 86)(47 130 142 88)(49 132 144 90)(51 134 146 92)(53 136 148 94)(55 138 150 96)(57 140 152 98)(59 122 154 100)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 135)(82 134)(83 133)(84 132)(85 131)(86 130)(87 129)(88 128)(89 127)(90 126)(91 125)(92 124)(93 123)(94 122)(95 121)(96 140)(97 139)(98 138)(99 137)(100 136)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111110104022448202020224420202···244448···84···4

47 irreducible representations

dim11111111222222222244444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5SD16D10D10D10D20C8.C22D4×D5D4×D5D5×SD16Q16⋊D5
kernelQ82D20C10.Q16D101C8C5×Q8⋊C4C4⋊D20C2×C40⋊C2C2×Q8⋊D5C2×Q8×D5Dic10C2×Dic5C5×Q8C22×D5Q8⋊C4D10C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps11111111212124222812244

Matrix representation of Q82D20 in GL4(𝔽41) generated by

1000
0100
0001
00400
,
1000
0100
001526
002626
,
303900
161400
0010
00040
,
1000
84000
0010
00040
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,15,26,0,0,26,26],[30,16,0,0,39,14,0,0,0,0,1,0,0,0,0,40],[1,8,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

Q82D20 in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_{20}
% in TeX

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

G:=SmallGroup(320,433);
// by ID

G=gap.SmallGroup(320,433);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,851,438,102,12550]);
// Polycyclic

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

׿
×
𝔽