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## G = (C2×C20)⋊5D4order 320 = 26·5

### 1st semidirect product of C2×C20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C20)⋊5D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×D20 — (C2×C20)⋊5D4
 Lower central C5 — C22×C10 — (C2×C20)⋊5D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C20)⋊5D4
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=ab-1, dcd=c-1 >

Subgroups: 1702 in 322 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C22×D4, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C232D4, D10⋊C4, C2×D20, C22×Dic5, C22×C20, C23×D5, C5×C2.C42, C2×D10⋊C4, C22×D20, (C2×C20)⋊5D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C41D4, D20, C22×D5, C232D4, C2×D20, D4×D5, Q82D5, C204D4, C22⋊D20, C4⋊D20, (C2×C20)⋊5D4

Smallest permutation representation of (C2×C20)⋊5D4
On 160 points
Generators in S160
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 156)(42 157)(43 158)(44 159)(45 160)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 110)(62 111)(63 112)(64 113)(65 114)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)
(1 53 104 28)(2 149 105 99)(3 55 106 30)(4 151 107 81)(5 57 108 32)(6 153 109 83)(7 59 110 34)(8 155 111 85)(9 41 112 36)(10 157 113 87)(11 43 114 38)(12 159 115 89)(13 45 116 40)(14 141 117 91)(15 47 118 22)(16 143 119 93)(17 49 120 24)(18 145 101 95)(19 51 102 26)(20 147 103 97)(21 128 46 68)(23 130 48 70)(25 132 50 72)(27 134 52 74)(29 136 54 76)(31 138 56 78)(33 140 58 80)(35 122 60 62)(37 124 42 64)(39 126 44 66)(61 84 121 154)(63 86 123 156)(65 88 125 158)(67 90 127 160)(69 92 129 142)(71 94 131 144)(73 96 133 146)(75 98 135 148)(77 100 137 150)(79 82 139 152)
(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 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 60)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 144)(82 143)(83 142)(84 141)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)(91 154)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4F 4G 4H 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 ··· 20 4 ··· 4 20 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D5 C4○D4 D10 D20 D4×D5 Q8⋊2D5 kernel (C2×C20)⋊5D4 C5×C2.C42 C2×D10⋊C4 C22×D20 C2×C20 C22×D5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 # reps 1 1 3 3 6 6 2 2 6 24 6 2

Matrix representation of (C2×C20)⋊5D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 0 0 0 0 1 0 0 0 0 0 37 40
,
 28 39 0 0 0 0 2 16 0 0 0 0 0 0 0 40 0 0 0 0 1 7 0 0 0 0 0 0 16 8 0 0 0 0 14 25
,
 28 39 0 0 0 0 2 13 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 16 8 0 0 0 0 4 25

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[28,2,0,0,0,0,39,16,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,16,14,0,0,0,0,8,25],[28,2,0,0,0,0,39,13,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,16,4,0,0,0,0,8,25] >;

(C2×C20)⋊5D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_5D_4
% in TeX

G:=Group("(C2xC20):5D4");
// GroupNames label

G:=SmallGroup(320,298);
// by ID

G=gap.SmallGroup(320,298);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,58,12550]);
// Polycyclic

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

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