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## G = C2×C80⋊C2order 320 = 26·5

### Direct product of C2 and C80⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C80⋊C2
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×C2×C8 — C2×C80⋊C2
 Lower central C5 — C10 — C2×C80⋊C2
 Upper central C1 — C2×C8 — C2×C16

Generators and relations for C2×C80⋊C2
G = < a,b,c | a2=b80=c2=1, ab=ba, ac=ca, cbc=b9 >

Subgroups: 238 in 90 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, C2×C16, M5(2), C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C52C16, C80, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, C80⋊C2, C2×C52C16, C2×C80, D5×C2×C8, C2×C80⋊C2
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, D10, M5(2), C22×C8, C4×D5, C22×D5, C2×M5(2), C8×D5, C2×C4×D5, C80⋊C2, D5×C2×C8, C2×C80⋊C2

Smallest permutation representation of C2×C80⋊C2
On 160 points
Generators in S160
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 157)(63 158)(64 159)(65 160)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)
(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)
(2 10)(3 19)(4 28)(5 37)(6 46)(7 55)(8 64)(9 73)(12 20)(13 29)(14 38)(15 47)(16 56)(17 65)(18 74)(22 30)(23 39)(24 48)(25 57)(26 66)(27 75)(32 40)(33 49)(34 58)(35 67)(36 76)(42 50)(43 59)(44 68)(45 77)(52 60)(53 69)(54 78)(62 70)(63 79)(72 80)(81 121)(82 130)(83 139)(84 148)(85 157)(87 95)(88 104)(89 113)(90 122)(91 131)(92 140)(93 149)(94 158)(97 105)(98 114)(99 123)(100 132)(101 141)(102 150)(103 159)(107 115)(108 124)(109 133)(110 142)(111 151)(112 160)(117 125)(118 134)(119 143)(120 152)(127 135)(128 144)(129 153)(137 145)(138 154)(147 155)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 8I 8J 8K 8L 10A ··· 10F 16A ··· 16H 16I ··· 16P 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 8 8 8 8 10 ··· 10 16 ··· 16 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 10 10 1 1 1 1 10 10 2 2 1 ··· 1 10 10 10 10 2 ··· 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D5 D10 D10 M5(2) C4×D5 C4×D5 C8×D5 C8×D5 C80⋊C2 kernel C2×C80⋊C2 C80⋊C2 C2×C5⋊2C16 C2×C80 D5×C2×C8 C8×D5 C2×C5⋊2C8 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 C2×C16 C16 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 2 4 2 8 4 4 8 8 32

Matrix representation of C2×C80⋊C2 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240
,
 110 64 0 0 177 0 0 0 0 0 188 240 0 0 1 137
,
 1 0 0 0 51 240 0 0 0 0 1 0 0 0 51 240
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[110,177,0,0,64,0,0,0,0,0,188,1,0,0,240,137],[1,51,0,0,0,240,0,0,0,0,1,51,0,0,0,240] >;

C2×C80⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{80}\rtimes C_2
% in TeX

G:=Group("C2xC80:C2");
// GroupNames label

G:=SmallGroup(320,527);
// by ID

G=gap.SmallGroup(320,527);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,58,80,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^80=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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