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G = C22⋊Dic20order 320 = 26·5

The semidirect product of C22 and Dic20 acting via Dic20/Dic10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C22⋊Dic20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C22×Dic10 — C22⋊Dic20
 Lower central C5 — C10 — C2×C20 — C22⋊Dic20
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for C22⋊Dic20
G = < a,b,c,d | a2=b2=c40=1, d2=c20, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 590 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C22⋊Q16, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C20.44D4, C5×C22⋊C8, C2×Dic20, C20.48D4, C22×Dic10, C22⋊Dic20
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C22≀C2, C2×Q16, C8.C22, D20, C22×D5, C22⋊Q16, Dic20, C2×D20, D4×D5, C22⋊D20, C2×Dic20, C8.D10, C22⋊Dic20

Smallest permutation representation of C22⋊Dic20
On 160 points
Generators in S160
(1 21)(2 87)(3 23)(4 89)(5 25)(6 91)(7 27)(8 93)(9 29)(10 95)(11 31)(12 97)(13 33)(14 99)(15 35)(16 101)(17 37)(18 103)(19 39)(20 105)(22 107)(24 109)(26 111)(28 113)(30 115)(32 117)(34 119)(36 81)(38 83)(40 85)(41 143)(42 62)(43 145)(44 64)(45 147)(46 66)(47 149)(48 68)(49 151)(50 70)(51 153)(52 72)(53 155)(54 74)(55 157)(56 76)(57 159)(58 78)(59 121)(60 80)(61 123)(63 125)(65 127)(67 129)(69 131)(71 133)(73 135)(75 137)(77 139)(79 141)(82 102)(84 104)(86 106)(88 108)(90 110)(92 112)(94 114)(96 116)(98 118)(100 120)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 121)(80 122)
(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)
(1 156 21 136)(2 155 22 135)(3 154 23 134)(4 153 24 133)(5 152 25 132)(6 151 26 131)(7 150 27 130)(8 149 28 129)(9 148 29 128)(10 147 30 127)(11 146 31 126)(12 145 32 125)(13 144 33 124)(14 143 34 123)(15 142 35 122)(16 141 36 121)(17 140 37 160)(18 139 38 159)(19 138 39 158)(20 137 40 157)(41 119 61 99)(42 118 62 98)(43 117 63 97)(44 116 64 96)(45 115 65 95)(46 114 66 94)(47 113 67 93)(48 112 68 92)(49 111 69 91)(50 110 70 90)(51 109 71 89)(52 108 72 88)(53 107 73 87)(54 106 74 86)(55 105 75 85)(56 104 76 84)(57 103 77 83)(58 102 78 82)(59 101 79 81)(60 100 80 120)

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 Q16 D10 D10 D20 D20 Dic20 C8.C22 D4×D5 C8.D10 kernel C22⋊Dic20 C20.44D4 C5×C22⋊C8 C2×Dic20 C20.48D4 C22×Dic10 Dic10 C2×C20 C22×C10 C22⋊C8 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C10 C4 C2 # reps 1 2 1 2 1 1 4 1 1 2 4 4 2 4 4 16 1 4 4

Matrix representation of C22⋊Dic20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 39 40 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 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 40 0 0 0 0 8 34 0 0 0 0 0 0 40 40 0 0 0 0 2 1 0 0 0 0 0 0 0 24 0 0 0 0 29 24
,
 3 5 0 0 0 0 23 38 0 0 0 0 0 0 40 0 0 0 0 0 2 1 0 0 0 0 0 0 24 35 0 0 0 0 21 17

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,0,0,29,0,0,0,0,24,24],[3,23,0,0,0,0,5,38,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,0,24,21,0,0,0,0,35,17] >;

C22⋊Dic20 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{20}
% in TeX

G:=Group("C2^2:Dic20");
// GroupNames label

G:=SmallGroup(320,366);
// by ID

G=gap.SmallGroup(320,366);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,226,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=c^20,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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