direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.F5, C10⋊1M5(2), C5⋊C16⋊1C22, (C4×D5).8C8, C5⋊1(C2×M5(2)), C8.32(C2×F5), (C2×C8).19F5, C20.24(C2×C8), C40.39(C2×C4), (C2×C40).23C4, (C8×D5).11C4, C4.19(D5⋊C8), D10.17(C2×C8), C10.9(C22×C8), (C22×D5).8C8, C4.46(C22×F5), C20.86(C22×C4), C5⋊2C8.36C23, Dic5.17(C2×C8), (C2×Dic5).13C8, (C8×D5).62C22, C22.13(D5⋊C8), (C2×C5⋊C16)⋊7C2, (C2×C4×D5).46C4, (D5×C2×C8).30C2, C2.10(C2×D5⋊C8), (C2×C10).11(C2×C8), C5⋊2C8.52(C2×C4), (C4×D5).92(C2×C4), (C2×C4).164(C2×F5), (C2×C20).173(C2×C4), (C2×C5⋊2C8).349C22, SmallGroup(320,1052)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8.F5
G = < a,b,c,d | a2=b8=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >
Subgroups: 250 in 90 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, M5(2), C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C8.F5, C2×C5⋊C16, D5×C2×C8, C2×C8.F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, M5(2), C22×C8, C2×F5, C2×M5(2), D5⋊C8, C22×F5, C8.F5, C2×D5⋊C8, C2×C8.F5
►On 160 points
Generators in S160
(1 140)(2 141)(3 142)(4 143)(5 144)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 112)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 159)(34 160)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 95)(50 96)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 113)(78 114)(79 115)(80 116)
(1 134 5 138 9 142 13 130)(2 143 6 131 10 135 14 139)(3 136 7 140 11 144 15 132)(4 129 8 133 12 137 16 141)(17 153 21 157 25 145 29 149)(18 146 22 150 26 154 30 158)(19 155 23 159 27 147 31 151)(20 148 24 152 28 156 32 160)(33 106 37 110 41 98 45 102)(34 99 38 103 42 107 46 111)(35 108 39 112 43 100 47 104)(36 101 40 105 44 109 48 97)(49 126 53 114 57 118 61 122)(50 119 54 123 58 127 62 115)(51 128 55 116 59 120 63 124)(52 121 56 125 60 113 64 117)(65 82 69 86 73 90 77 94)(66 91 70 95 74 83 78 87)(67 84 71 88 75 92 79 96)(68 93 72 81 76 85 80 89)
(1 60 29 45 67)(2 46 61 68 30)(3 69 47 31 62)(4 32 70 63 48)(5 64 17 33 71)(6 34 49 72 18)(7 73 35 19 50)(8 20 74 51 36)(9 52 21 37 75)(10 38 53 76 22)(11 77 39 23 54)(12 24 78 55 40)(13 56 25 41 79)(14 42 57 80 26)(15 65 43 27 58)(16 28 66 59 44)(81 146 131 99 126)(82 100 147 127 132)(83 128 101 133 148)(84 134 113 149 102)(85 150 135 103 114)(86 104 151 115 136)(87 116 105 137 152)(88 138 117 153 106)(89 154 139 107 118)(90 108 155 119 140)(91 120 109 141 156)(92 142 121 157 110)(93 158 143 111 122)(94 112 159 123 144)(95 124 97 129 160)(96 130 125 145 98)
(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)
G:=sub<Sym(160)| (1,140)(2,141)(3,142)(4,143)(5,144)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,112)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,159)(34,160)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,95)(50,96)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,113)(78,114)(79,115)(80,116), (1,134,5,138,9,142,13,130)(2,143,6,131,10,135,14,139)(3,136,7,140,11,144,15,132)(4,129,8,133,12,137,16,141)(17,153,21,157,25,145,29,149)(18,146,22,150,26,154,30,158)(19,155,23,159,27,147,31,151)(20,148,24,152,28,156,32,160)(33,106,37,110,41,98,45,102)(34,99,38,103,42,107,46,111)(35,108,39,112,43,100,47,104)(36,101,40,105,44,109,48,97)(49,126,53,114,57,118,61,122)(50,119,54,123,58,127,62,115)(51,128,55,116,59,120,63,124)(52,121,56,125,60,113,64,117)(65,82,69,86,73,90,77,94)(66,91,70,95,74,83,78,87)(67,84,71,88,75,92,79,96)(68,93,72,81,76,85,80,89), (1,60,29,45,67)(2,46,61,68,30)(3,69,47,31,62)(4,32,70,63,48)(5,64,17,33,71)(6,34,49,72,18)(7,73,35,19,50)(8,20,74,51,36)(9,52,21,37,75)(10,38,53,76,22)(11,77,39,23,54)(12,24,78,55,40)(13,56,25,41,79)(14,42,57,80,26)(15,65,43,27,58)(16,28,66,59,44)(81,146,131,99,126)(82,100,147,127,132)(83,128,101,133,148)(84,134,113,149,102)(85,150,135,103,114)(86,104,151,115,136)(87,116,105,137,152)(88,138,117,153,106)(89,154,139,107,118)(90,108,155,119,140)(91,120,109,141,156)(92,142,121,157,110)(93,158,143,111,122)(94,112,159,123,144)(95,124,97,129,160)(96,130,125,145,98), (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)>;
G:=Group( (1,140)(2,141)(3,142)(4,143)(5,144)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,112)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,159)(34,160)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,95)(50,96)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,113)(78,114)(79,115)(80,116), (1,134,5,138,9,142,13,130)(2,143,6,131,10,135,14,139)(3,136,7,140,11,144,15,132)(4,129,8,133,12,137,16,141)(17,153,21,157,25,145,29,149)(18,146,22,150,26,154,30,158)(19,155,23,159,27,147,31,151)(20,148,24,152,28,156,32,160)(33,106,37,110,41,98,45,102)(34,99,38,103,42,107,46,111)(35,108,39,112,43,100,47,104)(36,101,40,105,44,109,48,97)(49,126,53,114,57,118,61,122)(50,119,54,123,58,127,62,115)(51,128,55,116,59,120,63,124)(52,121,56,125,60,113,64,117)(65,82,69,86,73,90,77,94)(66,91,70,95,74,83,78,87)(67,84,71,88,75,92,79,96)(68,93,72,81,76,85,80,89), (1,60,29,45,67)(2,46,61,68,30)(3,69,47,31,62)(4,32,70,63,48)(5,64,17,33,71)(6,34,49,72,18)(7,73,35,19,50)(8,20,74,51,36)(9,52,21,37,75)(10,38,53,76,22)(11,77,39,23,54)(12,24,78,55,40)(13,56,25,41,79)(14,42,57,80,26)(15,65,43,27,58)(16,28,66,59,44)(81,146,131,99,126)(82,100,147,127,132)(83,128,101,133,148)(84,134,113,149,102)(85,150,135,103,114)(86,104,151,115,136)(87,116,105,137,152)(88,138,117,153,106)(89,154,139,107,118)(90,108,155,119,140)(91,120,109,141,156)(92,142,121,157,110)(93,158,143,111,122)(94,112,159,123,144)(95,124,97,129,160)(96,130,125,145,98), (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) );
G=PermutationGroup([[(1,140),(2,141),(3,142),(4,143),(5,144),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,112),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,159),(34,160),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,157),(48,158),(49,95),(50,96),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,113),(78,114),(79,115),(80,116)], [(1,134,5,138,9,142,13,130),(2,143,6,131,10,135,14,139),(3,136,7,140,11,144,15,132),(4,129,8,133,12,137,16,141),(17,153,21,157,25,145,29,149),(18,146,22,150,26,154,30,158),(19,155,23,159,27,147,31,151),(20,148,24,152,28,156,32,160),(33,106,37,110,41,98,45,102),(34,99,38,103,42,107,46,111),(35,108,39,112,43,100,47,104),(36,101,40,105,44,109,48,97),(49,126,53,114,57,118,61,122),(50,119,54,123,58,127,62,115),(51,128,55,116,59,120,63,124),(52,121,56,125,60,113,64,117),(65,82,69,86,73,90,77,94),(66,91,70,95,74,83,78,87),(67,84,71,88,75,92,79,96),(68,93,72,81,76,85,80,89)], [(1,60,29,45,67),(2,46,61,68,30),(3,69,47,31,62),(4,32,70,63,48),(5,64,17,33,71),(6,34,49,72,18),(7,73,35,19,50),(8,20,74,51,36),(9,52,21,37,75),(10,38,53,76,22),(11,77,39,23,54),(12,24,78,55,40),(13,56,25,41,79),(14,42,57,80,26),(15,65,43,27,58),(16,28,66,59,44),(81,146,131,99,126),(82,100,147,127,132),(83,128,101,133,148),(84,134,113,149,102),(85,150,135,103,114),(86,104,151,115,136),(87,116,105,137,152),(88,138,117,153,106),(89,154,139,107,118),(90,108,155,119,140),(91,120,109,141,156),(92,142,121,157,110),(93,158,143,111,122),(94,112,159,123,144),(95,124,97,129,160),(96,130,125,145,98)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | C8.F5 |
| kernel | C2×C8.F5 | C8.F5 | C2×C5⋊C16 | D5×C2×C8 | C8×D5 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×C8.F5 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 233 | 0 | 0 | 0 | 0 | 0 |
| 63 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 19 | 1 | 0 | 0 | 0 | 0 |
| 129 | 222 | 0 | 0 | 0 | 0 |
| 0 | 0 | 173 | 68 | 3 | 0 |
| 0 | 0 | 176 | 68 | 0 | 173 |
| 0 | 0 | 173 | 0 | 68 | 176 |
| 0 | 0 | 0 | 3 | 68 | 173 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[233,63,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[19,129,0,0,0,0,1,222,0,0,0,0,0,0,173,176,173,0,0,0,68,68,0,3,0,0,3,0,68,68,0,0,0,173,176,173] >;
C2×C8.F5 in GAP, Magma, Sage, TeX
C_2\times C_8.F_5
% in TeX
G:=Group("C2xC8.F5"); // GroupNames label
G:=SmallGroup(320,1052);
// by ID
G=gap.SmallGroup(320,1052);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations