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