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 79 25 125 144)(2 80 26 126 129)(3 65 27 127 130)(4 66 28 128 131)(5 67 29 113 132)(6 68 30 114 133)(7 69 31 115 134)(8 70 32 116 135)(9 71 17 117 136)(10 72 18 118 137)(11 73 19 119 138)(12 74 20 120 139)(13 75 21 121 140)(14 76 22 122 141)(15 77 23 123 142)(16 78 24 124 143)(33 94 110 145 57)(34 95 111 146 58)(35 96 112 147 59)(36 81 97 148 60)(37 82 98 149 61)(38 83 99 150 62)(39 84 100 151 63)(40 85 101 152 64)(41 86 102 153 49)(42 87 103 154 50)(43 88 104 155 51)(44 89 105 156 52)(45 90 106 157 53)(46 91 107 158 54)(47 92 108 159 55)(48 93 109 160 56)
(1 35 13 39 9 43 5 47)(2 34 14 38 10 42 6 46)(3 33 15 37 11 41 7 45)(4 48 16 36 12 40 8 44)(17 51 113 92 25 59 121 84)(18 50 114 91 26 58 122 83)(19 49 115 90 27 57 123 82)(20 64 116 89 28 56 124 81)(21 63 117 88 29 55 125 96)(22 62 118 87 30 54 126 95)(23 61 119 86 31 53 127 94)(24 60 120 85 32 52 128 93)(65 110 142 149 73 102 134 157)(66 109 143 148 74 101 135 156)(67 108 144 147 75 100 136 155)(68 107 129 146 76 99 137 154)(69 106 130 145 77 98 138 153)(70 105 131 160 78 97 139 152)(71 104 132 159 79 112 140 151)(72 103 133 158 80 111 141 150)
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), (1,79,25,125,144)(2,80,26,126,129)(3,65,27,127,130)(4,66,28,128,131)(5,67,29,113,132)(6,68,30,114,133)(7,69,31,115,134)(8,70,32,116,135)(9,71,17,117,136)(10,72,18,118,137)(11,73,19,119,138)(12,74,20,120,139)(13,75,21,121,140)(14,76,22,122,141)(15,77,23,123,142)(16,78,24,124,143)(33,94,110,145,57)(34,95,111,146,58)(35,96,112,147,59)(36,81,97,148,60)(37,82,98,149,61)(38,83,99,150,62)(39,84,100,151,63)(40,85,101,152,64)(41,86,102,153,49)(42,87,103,154,50)(43,88,104,155,51)(44,89,105,156,52)(45,90,106,157,53)(46,91,107,158,54)(47,92,108,159,55)(48,93,109,160,56), (1,35,13,39,9,43,5,47)(2,34,14,38,10,42,6,46)(3,33,15,37,11,41,7,45)(4,48,16,36,12,40,8,44)(17,51,113,92,25,59,121,84)(18,50,114,91,26,58,122,83)(19,49,115,90,27,57,123,82)(20,64,116,89,28,56,124,81)(21,63,117,88,29,55,125,96)(22,62,118,87,30,54,126,95)(23,61,119,86,31,53,127,94)(24,60,120,85,32,52,128,93)(65,110,142,149,73,102,134,157)(66,109,143,148,74,101,135,156)(67,108,144,147,75,100,136,155)(68,107,129,146,76,99,137,154)(69,106,130,145,77,98,138,153)(70,105,131,160,78,97,139,152)(71,104,132,159,79,112,140,151)(72,103,133,158,80,111,141,150)>;
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), (1,79,25,125,144)(2,80,26,126,129)(3,65,27,127,130)(4,66,28,128,131)(5,67,29,113,132)(6,68,30,114,133)(7,69,31,115,134)(8,70,32,116,135)(9,71,17,117,136)(10,72,18,118,137)(11,73,19,119,138)(12,74,20,120,139)(13,75,21,121,140)(14,76,22,122,141)(15,77,23,123,142)(16,78,24,124,143)(33,94,110,145,57)(34,95,111,146,58)(35,96,112,147,59)(36,81,97,148,60)(37,82,98,149,61)(38,83,99,150,62)(39,84,100,151,63)(40,85,101,152,64)(41,86,102,153,49)(42,87,103,154,50)(43,88,104,155,51)(44,89,105,156,52)(45,90,106,157,53)(46,91,107,158,54)(47,92,108,159,55)(48,93,109,160,56), (1,35,13,39,9,43,5,47)(2,34,14,38,10,42,6,46)(3,33,15,37,11,41,7,45)(4,48,16,36,12,40,8,44)(17,51,113,92,25,59,121,84)(18,50,114,91,26,58,122,83)(19,49,115,90,27,57,123,82)(20,64,116,89,28,56,124,81)(21,63,117,88,29,55,125,96)(22,62,118,87,30,54,126,95)(23,61,119,86,31,53,127,94)(24,60,120,85,32,52,128,93)(65,110,142,149,73,102,134,157)(66,109,143,148,74,101,135,156)(67,108,144,147,75,100,136,155)(68,107,129,146,76,99,137,154)(69,106,130,145,77,98,138,153)(70,105,131,160,78,97,139,152)(71,104,132,159,79,112,140,151)(72,103,133,158,80,111,141,150) );
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)], [(1,79,25,125,144),(2,80,26,126,129),(3,65,27,127,130),(4,66,28,128,131),(5,67,29,113,132),(6,68,30,114,133),(7,69,31,115,134),(8,70,32,116,135),(9,71,17,117,136),(10,72,18,118,137),(11,73,19,119,138),(12,74,20,120,139),(13,75,21,121,140),(14,76,22,122,141),(15,77,23,123,142),(16,78,24,124,143),(33,94,110,145,57),(34,95,111,146,58),(35,96,112,147,59),(36,81,97,148,60),(37,82,98,149,61),(38,83,99,150,62),(39,84,100,151,63),(40,85,101,152,64),(41,86,102,153,49),(42,87,103,154,50),(43,88,104,155,51),(44,89,105,156,52),(45,90,106,157,53),(46,91,107,158,54),(47,92,108,159,55),(48,93,109,160,56)], [(1,35,13,39,9,43,5,47),(2,34,14,38,10,42,6,46),(3,33,15,37,11,41,7,45),(4,48,16,36,12,40,8,44),(17,51,113,92,25,59,121,84),(18,50,114,91,26,58,122,83),(19,49,115,90,27,57,123,82),(20,64,116,89,28,56,124,81),(21,63,117,88,29,55,125,96),(22,62,118,87,30,54,126,95),(23,61,119,86,31,53,127,94),(24,60,120,85,32,52,128,93),(65,110,142,149,73,102,134,157),(66,109,143,148,74,101,135,156),(67,108,144,147,75,100,136,155),(68,107,129,146,76,99,137,154),(69,106,130,145,77,98,138,153),(70,105,131,160,78,97,139,152),(71,104,132,159,79,112,140,151),(72,103,133,158,80,111,141,150)]])
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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