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G = D20⋊Q8order 320 = 26·5

1st semidirect product of D20 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D201Q8, Dic5.14SD16, C20⋊Q86C2, C4.3(Q8×D5), C4.Q811D5, C53(D42Q8), C20.15(C2×Q8), C4⋊C4.164D10, (C2×C8).141D10, C2.25(D5×SD16), D206C4.5C2, D208C4.5C2, C4.76(C4○D20), C20.Q818C2, C20.8Q830C2, (C2×Dic5).52D4, C10.41(C2×SD16), C22.221(D4×D5), D205C4.13C2, C20.168(C4○D4), C2.24(D40⋊C2), C10.73(C8⋊C22), (C2×C40).288C22, (C2×C20).286C23, (C2×D20).82C22, C10.37(C22⋊Q8), C2.14(D10⋊Q8), C4⋊Dic5.114C22, (C4×Dic5).38C22, (C5×C4.Q8)⋊19C2, (C2×C10).291(C2×D4), (C5×C4⋊C4).79C22, (C2×C52C8).63C22, (C2×C4).389(C22×D5), SmallGroup(320,497)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20⋊Q8
C1C5C10C2×C10C2×C20C2×D20D208C4 — D20⋊Q8
C5C10C2×C20 — D20⋊Q8
C1C22C2×C4C4.Q8

Generators and relations for D20⋊Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a9, cbc-1=a3b, dbd-1=a18b, 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, C4.Q8, C4.Q8, 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, D42Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.Q8, D206C4, C20.8Q8, D205C4, C5×C4.Q8, C20⋊Q8, D208C4, D20⋊Q8
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, C22×D5, D42Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×SD16, D40⋊C2, D20⋊Q8

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim111111112222222244444
type++++++++-+++++-++
imageC1C2C2C2C2C2C2C2Q8D4D5SD16C4○D4D10D10C4○D20C8⋊C22Q8×D5D4×D5D5×SD16D40⋊C2
kernelD20⋊Q8C20.Q8D206C4C20.8Q8D205C4C5×C4.Q8C20⋊Q8D208C4D20C2×Dic5C4.Q8Dic5C20C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111112224242812244

Matrix representation of D20⋊Q8 in GL4(𝔽41) generated by

343300
1100
002810
002413
,
1800
04000
002810
001613
,
383700
23300
002527
003316
,
191900
92200
001331
001728
G:=sub<GL(4,GF(41))| [34,1,0,0,33,1,0,0,0,0,28,24,0,0,10,13],[1,0,0,0,8,40,0,0,0,0,28,16,0,0,10,13],[38,23,0,0,37,3,0,0,0,0,25,33,0,0,27,16],[19,9,0,0,19,22,0,0,0,0,13,17,0,0,31,28] >;

D20⋊Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes Q_8
% in TeX

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

G:=SmallGroup(320,497);
// by ID

G=gap.SmallGroup(320,497);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,100,1684,851,102,12550]);
// 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^3*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations

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