metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80.1C4, C16.1F5, D10.3Q16, Dic5.10D8, C4.5(C4⋊F5), C8.22(C2×F5), C5⋊2C8.8Q8, C5⋊2C16.3C4, C40.22(C2×C4), (C4×D5).76D4, (D5×C16).3C2, C20.12(C4⋊C4), C5⋊1(C8.4Q8), C2.6(D5.D8), C10.3(C2.D8), D10.Q8.1C2, (C8×D5).51C22, SmallGroup(320,189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16.F5
G = < a,b,c | a16=b5=1, c4=a8, ab=ba, cac-1=a-1, cbc-1=b3 >
Smallest permutation representation of C16.F5
►On 160 points
Generators in S160
(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)
(1 23 97 140 82)(2 24 98 141 83)(3 25 99 142 84)(4 26 100 143 85)(5 27 101 144 86)(6 28 102 129 87)(7 29 103 130 88)(8 30 104 131 89)(9 31 105 132 90)(10 32 106 133 91)(11 17 107 134 92)(12 18 108 135 93)(13 19 109 136 94)(14 20 110 137 95)(15 21 111 138 96)(16 22 112 139 81)(33 157 59 77 127)(34 158 60 78 128)(35 159 61 79 113)(36 160 62 80 114)(37 145 63 65 115)(38 146 64 66 116)(39 147 49 67 117)(40 148 50 68 118)(41 149 51 69 119)(42 150 52 70 120)(43 151 53 71 121)(44 152 54 72 122)(45 153 55 73 123)(46 154 56 74 124)(47 155 57 75 125)(48 156 58 76 126)
(1 70 13 74 9 78 5 66)(2 69 14 73 10 77 6 65)(3 68 15 72 11 76 7 80)(4 67 16 71 12 75 8 79)(17 48 88 160 25 40 96 152)(18 47 89 159 26 39 81 151)(19 46 90 158 27 38 82 150)(20 45 91 157 28 37 83 149)(21 44 92 156 29 36 84 148)(22 43 93 155 30 35 85 147)(23 42 94 154 31 34 86 146)(24 41 95 153 32 33 87 145)(49 139 121 108 57 131 113 100)(50 138 122 107 58 130 114 99)(51 137 123 106 59 129 115 98)(52 136 124 105 60 144 116 97)(53 135 125 104 61 143 117 112)(54 134 126 103 62 142 118 111)(55 133 127 102 63 141 119 110)(56 132 128 101 64 140 120 109)
G:=sub<Sym(160)| (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), (1,23,97,140,82)(2,24,98,141,83)(3,25,99,142,84)(4,26,100,143,85)(5,27,101,144,86)(6,28,102,129,87)(7,29,103,130,88)(8,30,104,131,89)(9,31,105,132,90)(10,32,106,133,91)(11,17,107,134,92)(12,18,108,135,93)(13,19,109,136,94)(14,20,110,137,95)(15,21,111,138,96)(16,22,112,139,81)(33,157,59,77,127)(34,158,60,78,128)(35,159,61,79,113)(36,160,62,80,114)(37,145,63,65,115)(38,146,64,66,116)(39,147,49,67,117)(40,148,50,68,118)(41,149,51,69,119)(42,150,52,70,120)(43,151,53,71,121)(44,152,54,72,122)(45,153,55,73,123)(46,154,56,74,124)(47,155,57,75,125)(48,156,58,76,126), (1,70,13,74,9,78,5,66)(2,69,14,73,10,77,6,65)(3,68,15,72,11,76,7,80)(4,67,16,71,12,75,8,79)(17,48,88,160,25,40,96,152)(18,47,89,159,26,39,81,151)(19,46,90,158,27,38,82,150)(20,45,91,157,28,37,83,149)(21,44,92,156,29,36,84,148)(22,43,93,155,30,35,85,147)(23,42,94,154,31,34,86,146)(24,41,95,153,32,33,87,145)(49,139,121,108,57,131,113,100)(50,138,122,107,58,130,114,99)(51,137,123,106,59,129,115,98)(52,136,124,105,60,144,116,97)(53,135,125,104,61,143,117,112)(54,134,126,103,62,142,118,111)(55,133,127,102,63,141,119,110)(56,132,128,101,64,140,120,109)>;
G:=Group( (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), (1,23,97,140,82)(2,24,98,141,83)(3,25,99,142,84)(4,26,100,143,85)(5,27,101,144,86)(6,28,102,129,87)(7,29,103,130,88)(8,30,104,131,89)(9,31,105,132,90)(10,32,106,133,91)(11,17,107,134,92)(12,18,108,135,93)(13,19,109,136,94)(14,20,110,137,95)(15,21,111,138,96)(16,22,112,139,81)(33,157,59,77,127)(34,158,60,78,128)(35,159,61,79,113)(36,160,62,80,114)(37,145,63,65,115)(38,146,64,66,116)(39,147,49,67,117)(40,148,50,68,118)(41,149,51,69,119)(42,150,52,70,120)(43,151,53,71,121)(44,152,54,72,122)(45,153,55,73,123)(46,154,56,74,124)(47,155,57,75,125)(48,156,58,76,126), (1,70,13,74,9,78,5,66)(2,69,14,73,10,77,6,65)(3,68,15,72,11,76,7,80)(4,67,16,71,12,75,8,79)(17,48,88,160,25,40,96,152)(18,47,89,159,26,39,81,151)(19,46,90,158,27,38,82,150)(20,45,91,157,28,37,83,149)(21,44,92,156,29,36,84,148)(22,43,93,155,30,35,85,147)(23,42,94,154,31,34,86,146)(24,41,95,153,32,33,87,145)(49,139,121,108,57,131,113,100)(50,138,122,107,58,130,114,99)(51,137,123,106,59,129,115,98)(52,136,124,105,60,144,116,97)(53,135,125,104,61,143,117,112)(54,134,126,103,62,142,118,111)(55,133,127,102,63,141,119,110)(56,132,128,101,64,140,120,109) );
G=PermutationGroup([(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)], [(1,23,97,140,82),(2,24,98,141,83),(3,25,99,142,84),(4,26,100,143,85),(5,27,101,144,86),(6,28,102,129,87),(7,29,103,130,88),(8,30,104,131,89),(9,31,105,132,90),(10,32,106,133,91),(11,17,107,134,92),(12,18,108,135,93),(13,19,109,136,94),(14,20,110,137,95),(15,21,111,138,96),(16,22,112,139,81),(33,157,59,77,127),(34,158,60,78,128),(35,159,61,79,113),(36,160,62,80,114),(37,145,63,65,115),(38,146,64,66,116),(39,147,49,67,117),(40,148,50,68,118),(41,149,51,69,119),(42,150,52,70,120),(43,151,53,71,121),(44,152,54,72,122),(45,153,55,73,123),(46,154,56,74,124),(47,155,57,75,125),(48,156,58,76,126)], [(1,70,13,74,9,78,5,66),(2,69,14,73,10,77,6,65),(3,68,15,72,11,76,7,80),(4,67,16,71,12,75,8,79),(17,48,88,160,25,40,96,152),(18,47,89,159,26,39,81,151),(19,46,90,158,27,38,82,150),(20,45,91,157,28,37,83,149),(21,44,92,156,29,36,84,148),(22,43,93,155,30,35,85,147),(23,42,94,154,31,34,86,146),(24,41,95,153,32,33,87,145),(49,139,121,108,57,131,113,100),(50,138,122,107,58,130,114,99),(51,137,123,106,59,129,115,98),(52,136,124,105,60,144,116,97),(53,135,125,104,61,143,117,112),(54,134,126,103,62,142,118,111),(55,133,127,102,63,141,119,110),(56,132,128,101,64,140,120,109)])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 2 | 5 | 5 | 4 | 2 | 2 | 10 | 10 | 40 | 40 | 40 | 40 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | Q8 | D4 | D8 | Q16 | C8.4Q8 | F5 | C2×F5 | C4⋊F5 | D5.D8 | C16.F5 |
| kernel | C16.F5 | D5×C16 | D10.Q8 | C5⋊2C16 | C80 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C16.F5 ►in GL4(𝔽241) generated by
| 44 | 0 | 0 | 48 |
| 0 | 44 | 193 | 193 |
| 0 | 0 | 126 | 0 |
| 0 | 0 | 0 | 126 |
| 189 | 240 | 0 | 219 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 188 |
| 0 | 0 | 1 | 52 |
| 143 | 103 | 96 | 152 |
| 152 | 103 | 96 | 213 |
| 209 | 110 | 35 | 89 |
| 174 | 186 | 103 | 201 |
G:=sub<GL(4,GF(241))| [44,0,0,0,0,44,0,0,0,193,126,0,48,193,0,126],[189,1,0,0,240,0,0,0,0,0,240,1,219,0,188,52],[143,152,209,174,103,103,110,186,96,96,35,103,152,213,89,201] >;
C16.F5 in GAP, Magma, Sage, TeX
C_{16}.F_5 % in TeX
G:=Group("C16.F5"); // GroupNames label
G:=SmallGroup(320,189);
// by ID
G=gap.SmallGroup(320,189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,675,192,1684,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=1,c^4=a^8,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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