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G = D202Q8order 320 = 26·5

2nd semidirect product of D20 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D202Q8, Dic5.15D8, C20⋊Q87C2, C4.7(Q8×D5), C2.D88D5, C2.15(D5×D8), C53(D4⋊Q8), (C2×C8).31D10, C10.31(C2×D8), C20.23(C2×Q8), C4⋊C4.173D10, D206C4.8C2, D208C4.8C2, C4.84(C4○D20), C10.D823C2, C20.8Q825C2, (C2×Dic5).59D4, C22.235(D4×D5), D205C4.10C2, C20.172(C4○D4), (C2×C20).306C23, (C2×C40).245C22, (C2×D20).90C22, C10.41(C22⋊Q8), C2.18(D10⋊Q8), C2.25(Q16⋊D5), C10.74(C8.C22), C4⋊Dic5.128C22, (C4×Dic5).44C22, (C5×C2.D8)⋊15C2, (C2×C10).311(C2×D4), (C5×C4⋊C4).99C22, (C2×C52C8).75C22, (C2×C4).409(C22×D5), SmallGroup(320,517)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D202Q8
C1C5C10C2×C10C2×C20C2×D20D208C4 — D202Q8
C5C10C2×C20 — D202Q8
C1C22C2×C4C2.D8

Generators and relations for D202Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a9, cbc-1=a13b, dbd-1=a8b, dcd-1=c-1 >

Subgroups: 502 in 108 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×3], Q8 [×2], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C2.D8, C2.D8, C4×D4, C4⋊Q8, C52C8, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, D4⋊Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C10.D8, D206C4, C20.8Q8, D205C4, C5×C2.D8, C20⋊Q8, D208C4, D202Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, D8 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×D8, C8.C22, C22×D5, D4⋊Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×D8, Q16⋊D5, D202Q8

Smallest permutation representation of D202Q8
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 124 110 30)(2 123 111 29)(3 122 112 28)(4 121 113 27)(5 140 114 26)(6 139 115 25)(7 138 116 24)(8 137 117 23)(9 136 118 22)(10 135 119 21)(11 134 120 40)(12 133 101 39)(13 132 102 38)(14 131 103 37)(15 130 104 36)(16 129 105 35)(17 128 106 34)(18 127 107 33)(19 126 108 32)(20 125 109 31)(41 142 94 63)(42 141 95 62)(43 160 96 61)(44 159 97 80)(45 158 98 79)(46 157 99 78)(47 156 100 77)(48 155 81 76)(49 154 82 75)(50 153 83 74)(51 152 84 73)(52 151 85 72)(53 150 86 71)(54 149 87 70)(55 148 88 69)(56 147 89 68)(57 146 90 67)(58 145 91 66)(59 144 92 65)(60 143 93 64)
(1 86 110 53)(2 95 111 42)(3 84 112 51)(4 93 113 60)(5 82 114 49)(6 91 115 58)(7 100 116 47)(8 89 117 56)(9 98 118 45)(10 87 119 54)(11 96 120 43)(12 85 101 52)(13 94 102 41)(14 83 103 50)(15 92 104 59)(16 81 105 48)(17 90 106 57)(18 99 107 46)(19 88 108 55)(20 97 109 44)(21 70 135 149)(22 79 136 158)(23 68 137 147)(24 77 138 156)(25 66 139 145)(26 75 140 154)(27 64 121 143)(28 73 122 152)(29 62 123 141)(30 71 124 150)(31 80 125 159)(32 69 126 148)(33 78 127 157)(34 67 128 146)(35 76 129 155)(36 65 130 144)(37 74 131 153)(38 63 132 142)(39 72 133 151)(40 61 134 160)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120202244810102040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++-+++++--++
imageC1C2C2C2C2C2C2C2Q8D4D5D8C4○D4D10D10C4○D20C8.C22Q8×D5D4×D5D5×D8Q16⋊D5
kernelD202Q8C10.D8D206C4C20.8Q8D205C4C5×C2.D8C20⋊Q8D208C4D20C2×Dic5C2.D8Dic5C20C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111112224242812244

Matrix representation of D202Q8 in GL4(𝔽41) generated by

0100
40000
003540
0010
,
0100
1000
004035
0001
,
291200
121200
002121
001820
,
1000
0100
002813
00913
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,35,1,0,0,40,0],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,35,1],[29,12,0,0,12,12,0,0,0,0,21,18,0,0,21,20],[1,0,0,0,0,1,0,0,0,0,28,9,0,0,13,13] >;

D202Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_2Q_8
% in TeX

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

G:=SmallGroup(320,517);
// by ID

G=gap.SmallGroup(320,517);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,268,1684,851,102,12550]);
// 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,d*a*d^-1=a^9,c*b*c^-1=a^13*b,d*b*d^-1=a^8*b,d*c*d^-1=c^-1>;
// generators/relations

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