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G = D207D4order 320 = 26·5

7th semidirect product of D20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D207D4, (C5×Q8)⋊6D4, C4.64(D4×D5), C202D47C2, C56(D4⋊D4), Q83(C5⋊D4), C20.49(C2×D4), (C2×SD16)⋊13D5, (C2×D4).74D10, (C2×C8).148D10, D205C436C2, D101C834C2, C10.59C22≀C2, (C10×SD16)⋊22C2, C10.64(C4○D8), Q8⋊Dic530C2, (C2×Q8).117D10, (C22×D5).45D4, C22.269(D4×D5), C2.29(D40⋊C2), C10.79(C8⋊C22), (C2×C40).295C22, (C2×C20).449C23, (C2×Dic5).240D4, (D4×C10).98C22, (Q8×C10).78C22, C2.27(C23⋊D10), (C2×D20).125C22, C4⋊Dic5.176C22, C2.30(SD163D5), (C2×D4⋊D5)⋊20C2, C4.44(C2×C5⋊D4), (C2×Q82D5)⋊2C2, (C2×C4×D5).53C22, (C2×C10).361(C2×D4), (C2×C4).538(C22×D5), (C2×C52C8).158C22, SmallGroup(320,799)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D207D4
C1C5C10C20C2×C20C2×C4×D5C202D4 — D207D4
C5C10C2×C20 — D207D4
C1C22C2×C4C2×SD16

Generators and relations for D207D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, dad=a11, cbc-1=a3b, dbd=a5b, dcd=c-1 >

Subgroups: 750 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×11], Q8 [×2], Q8, C23 [×3], D5 [×3], C10 [×3], C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, C40, C4×D5 [×6], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C22×C10, D4⋊D4, C2×C52C8, C4⋊Dic5, D4⋊D5 [×2], C23.D5, C2×C40, C5×SD16 [×2], C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5 [×4], C2×C5⋊D4, D4×C10, Q8×C10, D101C8, D205C4, Q8⋊Dic5, C2×D4⋊D5, C202D4, C10×SD16, C2×Q82D5, D207D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊D4, D4×D5 [×2], C2×C5⋊D4, D40⋊C2, SD163D5, C23⋊D10, D207D4

Smallest permutation representation of D207D4
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)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)
(1 157 52 93)(2 156 53 92)(3 155 54 91)(4 154 55 90)(5 153 56 89)(6 152 57 88)(7 151 58 87)(8 150 59 86)(9 149 60 85)(10 148 41 84)(11 147 42 83)(12 146 43 82)(13 145 44 81)(14 144 45 100)(15 143 46 99)(16 142 47 98)(17 141 48 97)(18 160 49 96)(19 159 50 95)(20 158 51 94)(21 105 69 122)(22 104 70 121)(23 103 71 140)(24 102 72 139)(25 101 73 138)(26 120 74 137)(27 119 75 136)(28 118 76 135)(29 117 77 134)(30 116 78 133)(31 115 79 132)(32 114 80 131)(33 113 61 130)(34 112 62 129)(35 111 63 128)(36 110 64 127)(37 109 65 126)(38 108 66 125)(39 107 67 124)(40 106 68 123)
(1 24)(2 35)(3 26)(4 37)(5 28)(6 39)(7 30)(8 21)(9 32)(10 23)(11 34)(12 25)(13 36)(14 27)(15 38)(16 29)(17 40)(18 31)(19 22)(20 33)(41 71)(42 62)(43 73)(44 64)(45 75)(46 66)(47 77)(48 68)(49 79)(50 70)(51 61)(52 72)(53 63)(54 74)(55 65)(56 76)(57 67)(58 78)(59 69)(60 80)(81 110)(82 101)(83 112)(84 103)(85 114)(86 105)(87 116)(88 107)(89 118)(90 109)(91 120)(92 111)(93 102)(94 113)(95 104)(96 115)(97 106)(98 117)(99 108)(100 119)(121 159)(122 150)(123 141)(124 152)(125 143)(126 154)(127 145)(128 156)(129 147)(130 158)(131 149)(132 160)(133 151)(134 142)(135 153)(136 144)(137 155)(138 146)(139 157)(140 148)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444455888810···1010101010202020202020202040···40
size111182020202244101040224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8C5⋊D4C8⋊C22D4×D5D4×D5D40⋊C2SD163D5
kernelD207D4D101C8D205C4Q8⋊Dic5C2×D4⋊D5C202D4C10×SD16C2×Q82D5D20C2×Dic5C5×Q8C22×D5C2×SD16C2×C8C2×D4C2×Q8C10Q8C10C4C22C2C2
# reps11111111212122224812244

Matrix representation of D207D4 in GL4(𝔽41) generated by

1200
404000
003440
0081
,
1200
04000
0011
00040
,
303000
261100
003820
00203
,
01700
29000
001735
00724
G:=sub<GL(4,GF(41))| [1,40,0,0,2,40,0,0,0,0,34,8,0,0,40,1],[1,0,0,0,2,40,0,0,0,0,1,0,0,0,1,40],[30,26,0,0,30,11,0,0,0,0,38,20,0,0,20,3],[0,29,0,0,17,0,0,0,0,0,17,7,0,0,35,24] >;

D207D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_7D_4
% in TeX

G:=Group("D20:7D4");
// GroupNames label

G:=SmallGroup(320,799);
// by ID

G=gap.SmallGroup(320,799);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^11,c*b*c^-1=a^3*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

׿
×
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