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)), (C2×C8).19F5, C8.32(C2×F5), (C2×C40).23C4, C40.39(C2×C4), C20.24(C2×C8), (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, (D5×C2×C8).30C2, (C2×C4×D5).46C4, 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
Subgroups: 250 in 90 conjugacy classes, 52 normal (22 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 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2) [×4], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16 [×4], C8×D5 [×4], C2×C5⋊2C8, C2×C40, C2×C4×D5, C8.F5 [×4], C2×C5⋊C16 [×2], D5×C2×C8, C2×C8.F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C2×F5 [×3], C2×M5(2), D5⋊C8 [×2], C22×F5, C8.F5 [×2], C2×D5⋊C8, C2×C8.F5
Generators and relations
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 >
►On 160 points
Generators in S160
(1 112)(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)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 81)(48 82)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 145)(125 146)(126 147)(127 148)(128 149)
(1 106 5 110 9 98 13 102)(2 99 6 103 10 107 14 111)(3 108 7 112 11 100 15 104)(4 101 8 105 12 109 16 97)(17 159 21 147 25 151 29 155)(18 152 22 156 26 160 30 148)(19 145 23 149 27 153 31 157)(20 154 24 158 28 146 32 150)(33 72 37 76 41 80 45 68)(34 65 38 69 42 73 46 77)(35 74 39 78 43 66 47 70)(36 67 40 71 44 75 48 79)(49 89 53 93 57 81 61 85)(50 82 54 86 58 90 62 94)(51 91 55 95 59 83 63 87)(52 84 56 88 60 92 64 96)(113 142 117 130 121 134 125 138)(114 135 118 139 122 143 126 131)(115 144 119 132 123 136 127 140)(116 137 120 141 124 129 128 133)
(1 137 61 41 147)(2 42 138 148 62)(3 149 43 63 139)(4 64 150 140 44)(5 141 49 45 151)(6 46 142 152 50)(7 153 47 51 143)(8 52 154 144 48)(9 129 53 33 155)(10 34 130 156 54)(11 157 35 55 131)(12 56 158 132 36)(13 133 57 37 159)(14 38 134 160 58)(15 145 39 59 135)(16 60 146 136 40)(17 98 128 93 72)(18 94 99 73 113)(19 74 95 114 100)(20 115 75 101 96)(21 102 116 81 76)(22 82 103 77 117)(23 78 83 118 104)(24 119 79 105 84)(25 106 120 85 80)(26 86 107 65 121)(27 66 87 122 108)(28 123 67 109 88)(29 110 124 89 68)(30 90 111 69 125)(31 70 91 126 112)(32 127 71 97 92)
(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)| (1,112)(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)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,81)(48,82)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149), (1,106,5,110,9,98,13,102)(2,99,6,103,10,107,14,111)(3,108,7,112,11,100,15,104)(4,101,8,105,12,109,16,97)(17,159,21,147,25,151,29,155)(18,152,22,156,26,160,30,148)(19,145,23,149,27,153,31,157)(20,154,24,158,28,146,32,150)(33,72,37,76,41,80,45,68)(34,65,38,69,42,73,46,77)(35,74,39,78,43,66,47,70)(36,67,40,71,44,75,48,79)(49,89,53,93,57,81,61,85)(50,82,54,86,58,90,62,94)(51,91,55,95,59,83,63,87)(52,84,56,88,60,92,64,96)(113,142,117,130,121,134,125,138)(114,135,118,139,122,143,126,131)(115,144,119,132,123,136,127,140)(116,137,120,141,124,129,128,133), (1,137,61,41,147)(2,42,138,148,62)(3,149,43,63,139)(4,64,150,140,44)(5,141,49,45,151)(6,46,142,152,50)(7,153,47,51,143)(8,52,154,144,48)(9,129,53,33,155)(10,34,130,156,54)(11,157,35,55,131)(12,56,158,132,36)(13,133,57,37,159)(14,38,134,160,58)(15,145,39,59,135)(16,60,146,136,40)(17,98,128,93,72)(18,94,99,73,113)(19,74,95,114,100)(20,115,75,101,96)(21,102,116,81,76)(22,82,103,77,117)(23,78,83,118,104)(24,119,79,105,84)(25,106,120,85,80)(26,86,107,65,121)(27,66,87,122,108)(28,123,67,109,88)(29,110,124,89,68)(30,90,111,69,125)(31,70,91,126,112)(32,127,71,97,92), (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( (1,112)(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)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,81)(48,82)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149), (1,106,5,110,9,98,13,102)(2,99,6,103,10,107,14,111)(3,108,7,112,11,100,15,104)(4,101,8,105,12,109,16,97)(17,159,21,147,25,151,29,155)(18,152,22,156,26,160,30,148)(19,145,23,149,27,153,31,157)(20,154,24,158,28,146,32,150)(33,72,37,76,41,80,45,68)(34,65,38,69,42,73,46,77)(35,74,39,78,43,66,47,70)(36,67,40,71,44,75,48,79)(49,89,53,93,57,81,61,85)(50,82,54,86,58,90,62,94)(51,91,55,95,59,83,63,87)(52,84,56,88,60,92,64,96)(113,142,117,130,121,134,125,138)(114,135,118,139,122,143,126,131)(115,144,119,132,123,136,127,140)(116,137,120,141,124,129,128,133), (1,137,61,41,147)(2,42,138,148,62)(3,149,43,63,139)(4,64,150,140,44)(5,141,49,45,151)(6,46,142,152,50)(7,153,47,51,143)(8,52,154,144,48)(9,129,53,33,155)(10,34,130,156,54)(11,157,35,55,131)(12,56,158,132,36)(13,133,57,37,159)(14,38,134,160,58)(15,145,39,59,135)(16,60,146,136,40)(17,98,128,93,72)(18,94,99,73,113)(19,74,95,114,100)(20,115,75,101,96)(21,102,116,81,76)(22,82,103,77,117)(23,78,83,118,104)(24,119,79,105,84)(25,106,120,85,80)(26,86,107,65,121)(27,66,87,122,108)(28,123,67,109,88)(29,110,124,89,68)(30,90,111,69,125)(31,70,91,126,112)(32,127,71,97,92), (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([(1,112),(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),(17,139),(18,140),(19,141),(20,142),(21,143),(22,144),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,81),(48,82),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,145),(125,146),(126,147),(127,148),(128,149)], [(1,106,5,110,9,98,13,102),(2,99,6,103,10,107,14,111),(3,108,7,112,11,100,15,104),(4,101,8,105,12,109,16,97),(17,159,21,147,25,151,29,155),(18,152,22,156,26,160,30,148),(19,145,23,149,27,153,31,157),(20,154,24,158,28,146,32,150),(33,72,37,76,41,80,45,68),(34,65,38,69,42,73,46,77),(35,74,39,78,43,66,47,70),(36,67,40,71,44,75,48,79),(49,89,53,93,57,81,61,85),(50,82,54,86,58,90,62,94),(51,91,55,95,59,83,63,87),(52,84,56,88,60,92,64,96),(113,142,117,130,121,134,125,138),(114,135,118,139,122,143,126,131),(115,144,119,132,123,136,127,140),(116,137,120,141,124,129,128,133)], [(1,137,61,41,147),(2,42,138,148,62),(3,149,43,63,139),(4,64,150,140,44),(5,141,49,45,151),(6,46,142,152,50),(7,153,47,51,143),(8,52,154,144,48),(9,129,53,33,155),(10,34,130,156,54),(11,157,35,55,131),(12,56,158,132,36),(13,133,57,37,159),(14,38,134,160,58),(15,145,39,59,135),(16,60,146,136,40),(17,98,128,93,72),(18,94,99,73,113),(19,74,95,114,100),(20,115,75,101,96),(21,102,116,81,76),(22,82,103,77,117),(23,78,83,118,104),(24,119,79,105,84),(25,106,120,85,80),(26,86,107,65,121),(27,66,87,122,108),(28,123,67,109,88),(29,110,124,89,68),(30,90,111,69,125),(31,70,91,126,112),(32,127,71,97,92)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀