p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C16⋊5C8, C2.2C82, C82.1C2, C4.9M5(2), (C2×C8).5C8, C8.27(C2×C8), C4.11(C4×C8), (C4×C8).12C4, (C2×C16).16C4, (C4×C16).15C2, (C2×C4).80C42, C22.12(C4×C8), C2.1(C16⋊5C4), C42.335(C2×C4), (C4×C8).439C22, (C2×C4).92(C2×C8), (C2×C8).256(C2×C4), SmallGroup(128,43)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊5C8
G = < a,b | a16=b8=1, bab-1=a9 >
Smallest permutation representation of C16⋊5C8
►Regular action on 128 points
Generators in S128
(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)
(1 124 84 62 70 34 32 109)(2 117 85 55 71 43 17 102)(3 126 86 64 72 36 18 111)(4 119 87 57 73 45 19 104)(5 128 88 50 74 38 20 97)(6 121 89 59 75 47 21 106)(7 114 90 52 76 40 22 99)(8 123 91 61 77 33 23 108)(9 116 92 54 78 42 24 101)(10 125 93 63 79 35 25 110)(11 118 94 56 80 44 26 103)(12 127 95 49 65 37 27 112)(13 120 96 58 66 46 28 105)(14 113 81 51 67 39 29 98)(15 122 82 60 68 48 30 107)(16 115 83 53 69 41 31 100)
G:=sub<Sym(128)| (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), (1,124,84,62,70,34,32,109)(2,117,85,55,71,43,17,102)(3,126,86,64,72,36,18,111)(4,119,87,57,73,45,19,104)(5,128,88,50,74,38,20,97)(6,121,89,59,75,47,21,106)(7,114,90,52,76,40,22,99)(8,123,91,61,77,33,23,108)(9,116,92,54,78,42,24,101)(10,125,93,63,79,35,25,110)(11,118,94,56,80,44,26,103)(12,127,95,49,65,37,27,112)(13,120,96,58,66,46,28,105)(14,113,81,51,67,39,29,98)(15,122,82,60,68,48,30,107)(16,115,83,53,69,41,31,100)>;
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), (1,124,84,62,70,34,32,109)(2,117,85,55,71,43,17,102)(3,126,86,64,72,36,18,111)(4,119,87,57,73,45,19,104)(5,128,88,50,74,38,20,97)(6,121,89,59,75,47,21,106)(7,114,90,52,76,40,22,99)(8,123,91,61,77,33,23,108)(9,116,92,54,78,42,24,101)(10,125,93,63,79,35,25,110)(11,118,94,56,80,44,26,103)(12,127,95,49,65,37,27,112)(13,120,96,58,66,46,28,105)(14,113,81,51,67,39,29,98)(15,122,82,60,68,48,30,107)(16,115,83,53,69,41,31,100) );
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)], [(1,124,84,62,70,34,32,109),(2,117,85,55,71,43,17,102),(3,126,86,64,72,36,18,111),(4,119,87,57,73,45,19,104),(5,128,88,50,74,38,20,97),(6,121,89,59,75,47,21,106),(7,114,90,52,76,40,22,99),(8,123,91,61,77,33,23,108),(9,116,92,54,78,42,24,101),(10,125,93,63,79,35,25,110),(11,118,94,56,80,44,26,103),(12,127,95,49,65,37,27,112),(13,120,96,58,66,46,28,105),(14,113,81,51,67,39,29,98),(15,122,82,60,68,48,30,107),(16,115,83,53,69,41,31,100)]])
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 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) |
| kernel | C16⋊5C8 | C82 | C4×C16 | C4×C8 | C2×C16 | C16 | C2×C8 | C4 |
| # reps | 1 | 1 | 2 | 4 | 8 | 32 | 16 | 16 |
Matrix representation of C16⋊5C8 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 10 | 16 |
| 0 | 13 | 7 |
| 9 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 13 | 0 |
G:=sub<GL(3,GF(17))| [13,0,0,0,10,13,0,16,7],[9,0,0,0,0,13,0,1,0] >;
C16⋊5C8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_5C_8 % in TeX
G:=Group("C16:5C8"); // GroupNames label
G:=SmallGroup(128,43);
// by ID
G=gap.SmallGroup(128,43);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,925,64,100,136,172]);
// Polycyclic
G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^9>;
// generators/relations
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