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G = C2×D20.2C4order 320 = 26·5

Direct product of C2 and D20.2C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D20.2C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×C4○D20 — C2×D20.2C4
 Lower central C5 — C10 — C2×D20.2C4
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2×D20.2C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >

Subgroups: 718 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×14], M4(2) [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×3], C2×M4(2), C2×M4(2) [×2], C8○D4 [×8], C2×C4○D4, C52C8 [×4], C40 [×4], Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×C8○D4, C8×D5 [×8], C8⋊D5 [×8], C2×C52C8 [×2], C2×C52C8 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, D5×C2×C8 [×2], C2×C8⋊D5 [×2], D20.2C4 [×8], C22×C52C8, C10×M4(2), C2×C4○D20, C2×D20.2C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C8○D4 [×2], C23×C4, C4×D5 [×4], C22×D5 [×7], C2×C8○D4, C2×C4×D5 [×6], C23×D5, D20.2C4 [×2], D5×C22×C4, C2×D20.2C4

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

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 D10 D10 D10 C8○D4 C4×D5 C4×D5 D20.2C4 kernel C2×D20.2C4 D5×C2×C8 C2×C8⋊D5 D20.2C4 C22×C5⋊2C8 C10×M4(2) C2×C4○D20 C2×Dic10 C2×D20 C4○D20 C2×C5⋊D4 C2×M4(2) C2×C8 M4(2) C22×C4 C10 C2×C4 C23 C2 # reps 1 2 2 8 1 1 1 2 2 8 4 2 4 8 2 8 12 4 8

Matrix representation of C2×D20.2C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 34 40 0 0 1 0 0 0 0 0 32 0 0 0 23 9
,
 7 1 0 0 34 34 0 0 0 0 40 1 0 0 0 1
,
 32 0 0 0 0 32 0 0 0 0 14 27 0 0 28 27
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,40,0,0,0,0,0,32,23,0,0,0,9],[7,34,0,0,1,34,0,0,0,0,40,0,0,0,1,1],[32,0,0,0,0,32,0,0,0,0,14,28,0,0,27,27] >;

C2×D20.2C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}._2C_4
% in TeX

G:=Group("C2xD20.2C4");
// GroupNames label

G:=SmallGroup(320,1416);
// by ID

G=gap.SmallGroup(320,1416);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,297,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations

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