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G = C2×C80⋊C2order 320 = 26·5

Direct product of C2 and C80⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C80⋊C2, C169D10, C8012C22, C103M5(2), C40.64C23, (C2×C16)⋊8D5, (C2×C80)⋊15C2, (C4×D5).3C8, (C8×D5).3C4, C8.44(C4×D5), C4.24(C8×D5), C54(C2×M5(2)), C20.63(C2×C8), C40.102(C2×C4), D10.10(C2×C8), (C2×C8).342D10, (C2×Dic5).6C8, (C22×D5).4C8, C8.58(C22×D5), C22.14(C8×D5), C52C1610C22, C10.37(C22×C8), Dic5.12(C2×C8), (C8×D5).45C22, C20.188(C22×C4), (C2×C40).409C22, C2.14(D5×C2×C8), (D5×C2×C8).19C2, (C2×C4×D5).24C4, C4.103(C2×C4×D5), (C2×C52C16)⋊11C2, (C2×C52C8).18C4, (C2×C10).43(C2×C8), C52C8.43(C2×C4), (C4×D5).81(C2×C4), (C2×C4).176(C4×D5), (C2×C20).422(C2×C4), SmallGroup(320,527)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C80⋊C2
C1C5C10C20C40C8×D5D5×C2×C8 — C2×C80⋊C2
C5C10 — C2×C80⋊C2
C1C2×C8C2×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 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C16 [×2], C16 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16, C2×C16, M5(2) [×4], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C52C16 [×2], C80 [×2], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C80⋊C2 [×4], C2×C52C16, C2×C80, D5×C2×C8, C2×C80⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, D10 [×3], M5(2) [×2], C22×C8, C4×D5 [×2], C22×D5, C2×M5(2), C8×D5 [×2], C2×C4×D5, C80⋊C2 [×2], D5×C2×C8, C2×C80⋊C2

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

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A···8H8I8J8K8L10A···10F16A···16H16I···16P20A···20H40A···40P80A···80AF
order122222444444558···8888810···1016···1616···1620···2040···4080···80
size1111101011111010221···1101010102···22···210···102···22···22···2

104 irreducible representations

dim11111111111222222222
type++++++++
imageC1C2C2C2C2C4C4C4C8C8C8D5D10D10M5(2)C4×D5C4×D5C8×D5C8×D5C80⋊C2
kernelC2×C80⋊C2C80⋊C2C2×C52C16C2×C80D5×C2×C8C8×D5C2×C52C8C2×C4×D5C4×D5C2×Dic5C22×D5C2×C16C16C2×C8C10C8C2×C4C4C22C2
# reps141114228442428448832

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

240000
024000
002400
000240
,
1106400
177000
00188240
001137
,
1000
5124000
0010
0051240
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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