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## G = C2×C20.55D4order 320 = 26·5

### Direct product of C2 and C20.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C20.55D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C20.55D4
 Lower central C5 — C10 — C2×C20.55D4
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C20.55D4
G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b15c3 >

Subgroups: 398 in 202 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C23×C4, C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C22⋊C8, C2×C52C8, C2×C52C8, C22×C20, C22×C20, C22×C20, C23×C10, C20.55D4, C22×C52C8, C23×C20, C2×C20.55D4
Quotients:

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

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 5A 5B 8A ··· 8P 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 D4 D5 M4(2) Dic5 D10 Dic5 C5⋊D4 C5⋊2C8 C4.Dic5 kernel C2×C20.55D4 C20.55D4 C22×C5⋊2C8 C23×C20 C22×C20 C23×C10 C22×C10 C2×C20 C23×C4 C2×C10 C22×C4 C22×C4 C24 C2×C4 C23 C22 # reps 1 4 2 1 6 2 16 4 2 4 6 6 2 16 16 16

Matrix representation of C2×C20.55D4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 1 0 0 0 0 31 0 0 0 0 4
,
 3 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
,
 38 0 0 0 0 1 0 0 0 0 0 1 0 0 40 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,1,0,0,0,0,31,0,0,0,0,4],[3,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[38,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0] >;

C2×C20.55D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{55}D_4
% in TeX

G:=Group("C2xC20.55D4");
// GroupNames label

G:=SmallGroup(320,833);
// by ID

G=gap.SmallGroup(320,833);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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