metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.6M5(2), C20.33M4(2), C22⋊(C5⋊C16), (C2×C10)⋊1C16, (C2×C20).4C8, C5⋊2(C22⋊C16), C5⋊2C8.54D4, C23.2(C5⋊C8), C10.10(C2×C16), (C22×C4).8F5, (C22×C10).3C8, C10.9(C22⋊C8), (C22×C20).30C4, C4.42(C22⋊F5), C2.3(C20.C8), C20.40(C22⋊C4), C4.11(C22.F5), C2.1(C23.2F5), (C2×C5⋊C16)⋊6C2, C2.5(C2×C5⋊C16), (C2×C4).3(C5⋊C8), C22.10(C2×C5⋊C8), (C2×C5⋊2C8).35C4, (C2×C10).26(C2×C8), (C2×C4).158(C2×F5), (C2×C20).164(C2×C4), (C22×C5⋊2C8).22C2, (C2×C5⋊2C8).347C22, SmallGroup(320,249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.6M5(2)
G = < a,b,c | a10=b16=c2=1, bab-1=a3, ac=ca, cbc=a5b9 >
Subgroups: 162 in 66 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, C22×C8, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C22×C10, C22⋊C16, C5⋊C16, C2×C5⋊2C8, C2×C5⋊2C8, C22×C20, C2×C5⋊C16, C22×C5⋊2C8, C10.6M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C16, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×C16, M5(2), C5⋊C8, C2×F5, C22⋊C16, C5⋊C16, C2×C5⋊C8, C22.F5, C22⋊F5, C2×C5⋊C16, C20.C8, C23.2F5, C10.6M5(2)
►On 160 points
Generators in S160
(1 96 113 80 48 19 155 55 143 97)(2 65 156 98 114 20 144 81 33 56)(3 99 129 57 157 21 34 66 115 82)(4 58 35 83 130 22 116 100 158 67)(5 84 117 68 36 23 159 59 131 101)(6 69 160 102 118 24 132 85 37 60)(7 103 133 61 145 25 38 70 119 86)(8 62 39 87 134 26 120 104 146 71)(9 88 121 72 40 27 147 63 135 105)(10 73 148 106 122 28 136 89 41 64)(11 107 137 49 149 29 42 74 123 90)(12 50 43 91 138 30 124 108 150 75)(13 92 125 76 44 31 151 51 139 109)(14 77 152 110 126 32 140 93 45 52)(15 111 141 53 153 17 46 78 127 94)(16 54 47 95 142 18 128 112 154 79)
(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 28)(4 30)(6 32)(8 18)(10 20)(12 22)(14 24)(16 26)(33 106)(35 108)(37 110)(39 112)(41 98)(43 100)(45 102)(47 104)(50 116)(52 118)(54 120)(56 122)(58 124)(60 126)(62 128)(64 114)(65 136)(67 138)(69 140)(71 142)(73 144)(75 130)(77 132)(79 134)(81 148)(83 150)(85 152)(87 154)(89 156)(91 158)(93 160)(95 146)
G:=sub<Sym(160)| (1,96,113,80,48,19,155,55,143,97)(2,65,156,98,114,20,144,81,33,56)(3,99,129,57,157,21,34,66,115,82)(4,58,35,83,130,22,116,100,158,67)(5,84,117,68,36,23,159,59,131,101)(6,69,160,102,118,24,132,85,37,60)(7,103,133,61,145,25,38,70,119,86)(8,62,39,87,134,26,120,104,146,71)(9,88,121,72,40,27,147,63,135,105)(10,73,148,106,122,28,136,89,41,64)(11,107,137,49,149,29,42,74,123,90)(12,50,43,91,138,30,124,108,150,75)(13,92,125,76,44,31,151,51,139,109)(14,77,152,110,126,32,140,93,45,52)(15,111,141,53,153,17,46,78,127,94)(16,54,47,95,142,18,128,112,154,79), (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), (2,28)(4,30)(6,32)(8,18)(10,20)(12,22)(14,24)(16,26)(33,106)(35,108)(37,110)(39,112)(41,98)(43,100)(45,102)(47,104)(50,116)(52,118)(54,120)(56,122)(58,124)(60,126)(62,128)(64,114)(65,136)(67,138)(69,140)(71,142)(73,144)(75,130)(77,132)(79,134)(81,148)(83,150)(85,152)(87,154)(89,156)(91,158)(93,160)(95,146)>;
G:=Group( (1,96,113,80,48,19,155,55,143,97)(2,65,156,98,114,20,144,81,33,56)(3,99,129,57,157,21,34,66,115,82)(4,58,35,83,130,22,116,100,158,67)(5,84,117,68,36,23,159,59,131,101)(6,69,160,102,118,24,132,85,37,60)(7,103,133,61,145,25,38,70,119,86)(8,62,39,87,134,26,120,104,146,71)(9,88,121,72,40,27,147,63,135,105)(10,73,148,106,122,28,136,89,41,64)(11,107,137,49,149,29,42,74,123,90)(12,50,43,91,138,30,124,108,150,75)(13,92,125,76,44,31,151,51,139,109)(14,77,152,110,126,32,140,93,45,52)(15,111,141,53,153,17,46,78,127,94)(16,54,47,95,142,18,128,112,154,79), (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), (2,28)(4,30)(6,32)(8,18)(10,20)(12,22)(14,24)(16,26)(33,106)(35,108)(37,110)(39,112)(41,98)(43,100)(45,102)(47,104)(50,116)(52,118)(54,120)(56,122)(58,124)(60,126)(62,128)(64,114)(65,136)(67,138)(69,140)(71,142)(73,144)(75,130)(77,132)(79,134)(81,148)(83,150)(85,152)(87,154)(89,156)(91,158)(93,160)(95,146) );
G=PermutationGroup([[(1,96,113,80,48,19,155,55,143,97),(2,65,156,98,114,20,144,81,33,56),(3,99,129,57,157,21,34,66,115,82),(4,58,35,83,130,22,116,100,158,67),(5,84,117,68,36,23,159,59,131,101),(6,69,160,102,118,24,132,85,37,60),(7,103,133,61,145,25,38,70,119,86),(8,62,39,87,134,26,120,104,146,71),(9,88,121,72,40,27,147,63,135,105),(10,73,148,106,122,28,136,89,41,64),(11,107,137,49,149,29,42,74,123,90),(12,50,43,91,138,30,124,108,150,75),(13,92,125,76,44,31,151,51,139,109),(14,77,152,110,126,32,140,93,45,52),(15,111,141,53,153,17,46,78,127,94),(16,54,47,95,142,18,128,112,154,79)], [(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)], [(2,28),(4,30),(6,32),(8,18),(10,20),(12,22),(14,24),(16,26),(33,106),(35,108),(37,110),(39,112),(41,98),(43,100),(45,102),(47,104),(50,116),(52,118),(54,120),(56,122),(58,124),(60,126),(62,128),(64,114),(65,136),(67,138),(69,140),(71,142),(73,144),(75,130),(77,132),(79,134),(81,148),(83,150),(85,152),(87,154),(89,156),(91,158),(93,160),(95,146)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10G | 16A | ··· | 16P | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | M4(2) | M5(2) | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C22.F5 | C22⋊F5 | C5⋊C16 | C20.C8 |
| kernel | C10.6M5(2) | C2×C5⋊C16 | C22×C5⋊2C8 | C2×C5⋊2C8 | C22×C20 | C2×C20 | C22×C10 | C2×C10 | C5⋊2C8 | C20 | C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C10.6M5(2) ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 189 | 52 | 0 | 0 |
| 0 | 0 | 196 | 45 | 0 | 240 |
| 0 | 0 | 115 | 160 | 1 | 190 |
| 74 | 4 | 0 | 0 | 0 | 0 |
| 137 | 167 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 81 | 126 | 239 | 51 |
| 0 | 0 | 230 | 169 | 115 | 0 |
| 0 | 0 | 64 | 53 | 115 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 204 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,189,196,115,0,0,51,52,45,160,0,0,0,0,0,1,0,0,0,0,240,190],[74,137,0,0,0,0,4,167,0,0,0,0,0,0,0,81,230,64,0,0,0,126,169,53,0,0,240,239,115,115,0,0,1,51,0,0],[1,204,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C10.6M5(2) in GAP, Magma, Sage, TeX
C_{10}._6M_5(2) % in TeX
G:=Group("C10.6M5(2)"); // GroupNames label
G:=SmallGroup(320,249);
// by ID
G=gap.SmallGroup(320,249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=a^5*b^9>;
// generators/relations