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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 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×2], C22, C22 [×10], C22 [×12], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C2×C8 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C20 [×4], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C22⋊C8 [×4], C22×C8 [×2], C23×C4, C52C8 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊C8, C2×C52C8 [×4], C2×C52C8 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C20.55D4 [×4], C22×C52C8 [×2], C23×C20, C2×C20.55D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C52C8 [×4], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C22⋊C8, C2×C52C8 [×6], C4.Dic5 [×2], C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C20.55D4 [×4], C22×C52C8, C2×C4.Dic5, C2×C23.D5, C2×C20.55D4

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

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

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

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

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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