Copied to
clipboard

G = D204D4order 320 = 26·5

4th semidirect product of D20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D204D4, Q83D20, (C5×Q8)⋊2D4, (C2×D40)⋊6C2, C4.8(C2×D20), C4.95(D4×D5), C4⋊D205C2, C53(D4⋊D4), C4⋊C4.27D10, Q8⋊C46D5, (C2×C8).18D10, D101C88C2, C20.124(C2×D4), D206C412C2, C10.26C22≀C2, C10.70(C4○D8), (C2×C40).18C22, (C2×Q8).110D10, (C22×D5).26D4, C22.201(D4×D5), C2.9(Q8.D10), C2.17(D40⋊C2), C10.63(C8⋊C22), (C2×C20).251C23, (C2×Dic5).211D4, (C2×D20).69C22, (Q8×C10).34C22, C2.29(C22⋊D20), (C2×Q8⋊D5)⋊4C2, (C5×Q8⋊C4)⋊6C2, (C2×Q82D5)⋊1C2, (C2×C4×D5).28C22, (C2×C10).264(C2×D4), (C5×C4⋊C4).52C22, (C2×C52C8).42C22, (C2×C4).358(C22×D5), SmallGroup(320,438)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D204D4
C1C5C10C20C2×C20C2×C4×D5C4⋊D20 — D204D4
C5C10C2×C20 — D204D4
C1C22C2×C4Q8⋊C4

Generators and relations for D204D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a11, cbc-1=a5b, dbd=a3b, dcd=c-1 >

Subgroups: 846 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 [×4], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5, C20 [×2], C20 [×3], D10 [×10], C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, C40, C4×D5 [×6], D20 [×2], D20 [×9], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×2], D4⋊D4, D40 [×2], C2×C52C8, D10⋊C4, Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20 [×2], C2×D20 [×2], Q82D5 [×4], Q8×C10, D206C4, D101C8, C5×Q8⋊C4, C4⋊D20, C2×D40, C2×Q8⋊D5, C2×Q82D5, D204D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, D20 [×2], C22×D5, D4⋊D4, C2×D20, D4×D5 [×2], C22⋊D20, D40⋊C2, Q8.D10, D204D4

Smallest permutation representation of D204D4
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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 140)(9 139)(10 138)(11 137)(12 136)(13 135)(14 134)(15 133)(16 132)(17 131)(18 130)(19 129)(20 128)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)
(1 148 70 58)(2 159 71 49)(3 150 72 60)(4 141 73 51)(5 152 74 42)(6 143 75 53)(7 154 76 44)(8 145 77 55)(9 156 78 46)(10 147 79 57)(11 158 80 48)(12 149 61 59)(13 160 62 50)(14 151 63 41)(15 142 64 52)(16 153 65 43)(17 144 66 54)(18 155 67 45)(19 146 68 56)(20 157 69 47)(21 126 106 100)(22 137 107 91)(23 128 108 82)(24 139 109 93)(25 130 110 84)(26 121 111 95)(27 132 112 86)(28 123 113 97)(29 134 114 88)(30 125 115 99)(31 136 116 90)(32 127 117 81)(33 138 118 92)(34 129 119 83)(35 140 120 94)(36 131 101 85)(37 122 102 96)(38 133 103 87)(39 124 104 98)(40 135 105 89)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 120)(37 119)(38 118)(39 117)(40 116)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444455888810···102020202020···2040···40
size111120202040224481010224420202···244448···84···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8D20C8⋊C22D4×D5D4×D5D40⋊C2Q8.D10
kernelD204D4D206C4D101C8C5×Q8⋊C4C4⋊D20C2×D40C2×Q8⋊D5C2×Q82D5D20C2×Dic5C5×Q8C22×D5Q8⋊C4C4⋊C4C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps11111111212122224812244

Matrix representation of D204D4 in GL4(𝔽41) generated by

13900
14000
00040
0017
,
01700
29000
003230
00119
,
9000
93200
003032
00911
,
40200
0100
0010
003440
G:=sub<GL(4,GF(41))| [1,1,0,0,39,40,0,0,0,0,0,1,0,0,40,7],[0,29,0,0,17,0,0,0,0,0,32,11,0,0,30,9],[9,9,0,0,0,32,0,0,0,0,30,9,0,0,32,11],[40,0,0,0,2,1,0,0,0,0,1,34,0,0,0,40] >;

D204D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_4D_4
% in TeX

G:=Group("D20:4D4");
// GroupNames label

G:=SmallGroup(320,438);
// by ID

G=gap.SmallGroup(320,438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,135,184,570,297,136,12550]);
// Polycyclic

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

׿
×
𝔽