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## G = C2×C4⋊D20order 320 = 26·5

### Direct product of C2 and C4⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C4⋊D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×C4⋊D20
 Lower central C5 — C2×C10 — C2×C4⋊D20
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C4⋊D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1902 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×36], C5, C2×C4 [×10], C2×C4 [×16], D4 [×24], C23, C23 [×26], D5 [×8], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4 [×24], C24 [×3], Dic5 [×2], C20 [×4], C20 [×4], D10 [×4], D10 [×32], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], C4×D5 [×8], D20 [×24], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×10], C2×C20 [×4], C22×D5 [×10], C22×D5 [×16], C22×C10, C2×C4⋊D4, D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C2×D20 [×12], C2×D20 [×12], C22×Dic5, C22×C20, C22×C20 [×2], C23×D5, C23×D5 [×2], C4⋊D20 [×8], C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, C22×D20, C22×D20 [×2], C2×C4⋊D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D20 [×4], C22×D5 [×7], C2×C4⋊D4, C2×D20 [×6], D4×D5 [×2], Q82D5 [×2], C23×D5, C4⋊D20 [×4], C22×D20, C2×D4×D5, C2×Q82D5, C2×C4⋊D20

Smallest permutation representation of C2×C4⋊D20
On 160 points
Generators in S160
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 81)(40 82)(41 120)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)
(1 63 34 115)(2 116 35 64)(3 65 36 117)(4 118 37 66)(5 67 38 119)(6 120 39 68)(7 69 40 101)(8 102 21 70)(9 71 22 103)(10 104 23 72)(11 73 24 105)(12 106 25 74)(13 75 26 107)(14 108 27 76)(15 77 28 109)(16 110 29 78)(17 79 30 111)(18 112 31 80)(19 61 32 113)(20 114 33 62)(41 81 121 146)(42 147 122 82)(43 83 123 148)(44 149 124 84)(45 85 125 150)(46 151 126 86)(47 87 127 152)(48 153 128 88)(49 89 129 154)(50 155 130 90)(51 91 131 156)(52 157 132 92)(53 93 133 158)(54 159 134 94)(55 95 135 160)(56 141 136 96)(57 97 137 142)(58 143 138 98)(59 99 139 144)(60 145 140 100)
(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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 104)(42 103)(43 102)(44 101)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 132)(62 131)(63 130)(64 129)(65 128)(66 127)(67 126)(68 125)(69 124)(70 123)(71 122)(72 121)(73 140)(74 139)(75 138)(76 137)(77 136)(78 135)(79 134)(80 133)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 20 20 20 20 2 2 2 2 4 4 4 4 10 10 10 10 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D20 D4×D5 Q8⋊2D5 kernel C2×C4⋊D20 C4⋊D20 C2×D10⋊C4 C10×C4⋊C4 D5×C22×C4 C22×D20 C2×C20 C22×D5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 2 1 1 3 4 4 2 4 8 6 16 4 4

Matrix representation of C2×C4⋊D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 29 36 0 0 0 0 29 12
,
 34 40 0 0 0 0 1 0 0 0 0 0 0 0 25 39 0 0 0 0 2 13 0 0 0 0 0 0 15 37 0 0 0 0 36 26
,
 7 1 0 0 0 0 34 34 0 0 0 0 0 0 35 1 0 0 0 0 6 6 0 0 0 0 0 0 1 0 0 0 0 0 28 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,29,0,0,0,0,36,12],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,15,36,0,0,0,0,37,26],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40] >;

C2×C4⋊D20 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_{20}
% in TeX

G:=Group("C2xC4:D20");
// GroupNames label

G:=SmallGroup(320,1178);
// by ID

G=gap.SmallGroup(320,1178);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,12550]);
// Polycyclic

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

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