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## G = C2×C20⋊D4order 320 = 26·5

### Direct product of C2 and C20⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C20⋊D4
 Lower central C5 — C2×C10 — C2×C20⋊D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C20⋊D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=c-1 >

Subgroups: 2014 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×40], C5, C2×C4 [×6], C2×C4 [×12], D4 [×48], C23, C23 [×4], C23 [×28], D5 [×4], C10, C10 [×6], C10 [×4], C42 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×44], C24 [×2], C24 [×2], Dic5 [×8], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C42, C41D4 [×8], C22×D4, C22×D4 [×5], D20 [×8], C2×Dic5 [×12], C5⋊D4 [×32], C2×C20 [×6], C5×D4 [×8], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C41D4, C4×Dic5 [×4], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×16], C2×C5⋊D4 [×16], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5 [×2], C23×C10 [×2], C2×C4×Dic5, C20⋊D4 [×8], C22×D20, C22×C5⋊D4 [×4], D4×C2×C10, C2×C20⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C41D4 [×4], C22×D4 [×3], C5⋊D4 [×4], C22×D5 [×7], C2×C41D4, D4×D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20⋊D4 [×4], C2×D4×D5 [×2], C22×C5⋊D4, C2×C20⋊D4

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C5⋊D4 D4×D5 kernel C2×C20⋊D4 C2×C4×Dic5 C20⋊D4 C22×D20 C22×C5⋊D4 D4×C2×C10 C2×Dic5 C2×C20 C22×D4 C22×C4 C2×D4 C24 C2×C4 C22 # reps 1 1 8 1 4 1 8 4 2 2 8 4 16 8

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

 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 0 0 40 2 0 0 0 0 40 1
,
 25 32 0 0 0 0 24 16 0 0 0 0 0 0 21 17 0 0 0 0 15 20 0 0 0 0 0 0 1 39 0 0 0 0 1 40
,
 40 0 0 0 0 0 40 1 0 0 0 0 0 0 6 1 0 0 0 0 6 35 0 0 0 0 0 0 1 0 0 0 0 0 1 40

G:=sub<GL(6,GF(41))| [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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[25,24,0,0,0,0,32,16,0,0,0,0,0,0,21,15,0,0,0,0,17,20,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C2×C20⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes D_4
% in TeX

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

G:=SmallGroup(320,1475);
// by ID

G=gap.SmallGroup(320,1475);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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