direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C80⋊C2, C16⋊9D10, C80⋊12C22, C10⋊3M5(2), C40.64C23, (C2×C16)⋊8D5, (C2×C80)⋊15C2, (C4×D5).3C8, (C8×D5).3C4, C8.44(C4×D5), C4.24(C8×D5), C5⋊4(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), C5⋊2C16⋊10C22, 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×C5⋊2C16)⋊11C2, (C2×C5⋊2C8).18C4, (C2×C10).43(C2×C8), C5⋊2C8.43(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
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, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊2C16, C80, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C80⋊C2, C2×C5⋊2C16, 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
►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