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## G = D20⋊4D4order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊4D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C4⋊D20 — D20⋊4D4
 Lower central C5 — C10 — C2×C20 — D20⋊4D4
 Upper central C1 — C22 — C2×C4 — Q8⋊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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, D4⋊D4, D40, C2×C52C8, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5, Q8×C10, D206C4, D101C8, C5×Q8⋊C4, C4⋊D20, C2×D40, C2×Q8⋊D5, C2×Q82D5, D204D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, D20, C22×D5, D4⋊D4, C2×D20, D4×D5, 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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 120)(21 83)(22 82)(23 81)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(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)(61 129)(62 128)(63 127)(64 126)(65 125)(66 124)(67 123)(68 122)(69 121)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)
(1 84 124 155)(2 95 125 146)(3 86 126 157)(4 97 127 148)(5 88 128 159)(6 99 129 150)(7 90 130 141)(8 81 131 152)(9 92 132 143)(10 83 133 154)(11 94 134 145)(12 85 135 156)(13 96 136 147)(14 87 137 158)(15 98 138 149)(16 89 139 160)(17 100 140 151)(18 91 121 142)(19 82 122 153)(20 93 123 144)(21 72 52 105)(22 63 53 116)(23 74 54 107)(24 65 55 118)(25 76 56 109)(26 67 57 120)(27 78 58 111)(28 69 59 102)(29 80 60 113)(30 71 41 104)(31 62 42 115)(32 73 43 106)(33 64 44 117)(34 75 45 108)(35 66 46 119)(36 77 47 110)(37 68 48 101)(38 79 49 112)(39 70 50 103)(40 61 51 114)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 160)(100 159)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 20 40 2 2 4 4 8 10 10 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D10 D10 D10 C4○D8 D20 C8⋊C22 D4×D5 D4×D5 D40⋊C2 Q8.D10 kernel D20⋊4D4 D20⋊6C4 D10⋊1C8 C5×Q8⋊C4 C4⋊D20 C2×D40 C2×Q8⋊D5 C2×Q8⋊2D5 D20 C2×Dic5 C5×Q8 C22×D5 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 C10 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of D204D4 in GL4(𝔽41) generated by

 1 39 0 0 1 40 0 0 0 0 0 40 0 0 1 7
,
 0 17 0 0 29 0 0 0 0 0 32 30 0 0 11 9
,
 9 0 0 0 9 32 0 0 0 0 30 32 0 0 9 11
,
 40 2 0 0 0 1 0 0 0 0 1 0 0 0 34 40
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

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