p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C8⋊3C16, C82.9C2, C8.24M4(2), C4.10M5(2), (C2×C8).6C8, C2.1(C4×C16), (C2×C16).7C4, (C4×C16).1C2, (C4×C8).13C4, C4.11(C2×C16), C2.2(C8⋊C8), (C2×C4).81C42, C22.13(C4×C8), C4.13(C8⋊C4), C2.2(C16⋊5C4), C42.336(C2×C4), (C4×C8).440C22, (C2×C4).93(C2×C8), (C2×C8).257(C2×C4), SmallGroup(128,44)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊C16
G = < a,b | a8=b16=1, bab-1=a5 >
Smallest permutation representation of C8⋊C16
►Regular action on 128 points
Generators in S128
(1 126 32 88 44 76 64 112)(2 77 17 97 45 127 49 89)(3 128 18 90 46 78 50 98)(4 79 19 99 47 113 51 91)(5 114 20 92 48 80 52 100)(6 65 21 101 33 115 53 93)(7 116 22 94 34 66 54 102)(8 67 23 103 35 117 55 95)(9 118 24 96 36 68 56 104)(10 69 25 105 37 119 57 81)(11 120 26 82 38 70 58 106)(12 71 27 107 39 121 59 83)(13 122 28 84 40 72 60 108)(14 73 29 109 41 123 61 85)(15 124 30 86 42 74 62 110)(16 75 31 111 43 125 63 87)
(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)
G:=sub<Sym(128)| (1,126,32,88,44,76,64,112)(2,77,17,97,45,127,49,89)(3,128,18,90,46,78,50,98)(4,79,19,99,47,113,51,91)(5,114,20,92,48,80,52,100)(6,65,21,101,33,115,53,93)(7,116,22,94,34,66,54,102)(8,67,23,103,35,117,55,95)(9,118,24,96,36,68,56,104)(10,69,25,105,37,119,57,81)(11,120,26,82,38,70,58,106)(12,71,27,107,39,121,59,83)(13,122,28,84,40,72,60,108)(14,73,29,109,41,123,61,85)(15,124,30,86,42,74,62,110)(16,75,31,111,43,125,63,87), (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)>;
G:=Group( (1,126,32,88,44,76,64,112)(2,77,17,97,45,127,49,89)(3,128,18,90,46,78,50,98)(4,79,19,99,47,113,51,91)(5,114,20,92,48,80,52,100)(6,65,21,101,33,115,53,93)(7,116,22,94,34,66,54,102)(8,67,23,103,35,117,55,95)(9,118,24,96,36,68,56,104)(10,69,25,105,37,119,57,81)(11,120,26,82,38,70,58,106)(12,71,27,107,39,121,59,83)(13,122,28,84,40,72,60,108)(14,73,29,109,41,123,61,85)(15,124,30,86,42,74,62,110)(16,75,31,111,43,125,63,87), (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) );
G=PermutationGroup([[(1,126,32,88,44,76,64,112),(2,77,17,97,45,127,49,89),(3,128,18,90,46,78,50,98),(4,79,19,99,47,113,51,91),(5,114,20,92,48,80,52,100),(6,65,21,101,33,115,53,93),(7,116,22,94,34,66,54,102),(8,67,23,103,35,117,55,95),(9,118,24,96,36,68,56,104),(10,69,25,105,37,119,57,81),(11,120,26,82,38,70,58,106),(12,71,27,107,39,121,59,83),(13,122,28,84,40,72,60,108),(14,73,29,109,41,123,61,85),(15,124,30,86,42,74,62,110),(16,75,31,111,43,125,63,87)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P | 8Q | ··· | 8AF | 16A | ··· | 16AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | M4(2) | M5(2) |
| kernel | C8⋊C16 | C82 | C4×C16 | C4×C8 | C2×C16 | C2×C8 | C8 | C8 | C4 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 8 | 8 |
Matrix representation of C8⋊C16 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 0 | 13 |
| 0 | 1 | 0 |
| 3 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 7 |
G:=sub<GL(3,GF(17))| [1,0,0,0,0,1,0,13,0],[3,0,0,0,10,0,0,0,7] >;
C8⋊C16 in GAP, Magma, Sage, TeX
C_8\rtimes C_{16} % in TeX
G:=Group("C8:C16"); // GroupNames label
G:=SmallGroup(128,44);
// by ID
G=gap.SmallGroup(128,44);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,253,64,100,136,124]);
// Polycyclic
G:=Group<a,b|a^8=b^16=1,b*a*b^-1=a^5>;
// generators/relations
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