p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C16⋊1C8, C42.41D4, C8.17M4(2), C4.3(C4⋊C8), (C2×C16).1C4, C8.19(C2×C8), (C2×C8).25Q8, C16⋊5C4.3C2, C8⋊2C8.12C2, C2.3(C8⋊2C8), C2.1(C8.Q8), (C2×C4).86SD16, C4.2(C8.C4), (C4×C8).284C22, C22.12(C4.Q8), (C2×C4).99(C4⋊C4), (C2×C8).229(C2×C4), SmallGroup(128,100)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊1C8
G = < a,b | a16=b8=1, bab-1=a11 >
Smallest permutation representation of C16⋊1C8
►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 80 33 31 121 85 49 108)(2 67 42 26 122 88 58 103)(3 70 35 21 123 91 51 98)(4 73 44 32 124 94 60 109)(5 76 37 27 125 81 53 104)(6 79 46 22 126 84 62 99)(7 66 39 17 127 87 55 110)(8 69 48 28 128 90 64 105)(9 72 41 23 113 93 57 100)(10 75 34 18 114 96 50 111)(11 78 43 29 115 83 59 106)(12 65 36 24 116 86 52 101)(13 68 45 19 117 89 61 112)(14 71 38 30 118 92 54 107)(15 74 47 25 119 95 63 102)(16 77 40 20 120 82 56 97)
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), (1,80,33,31,121,85,49,108)(2,67,42,26,122,88,58,103)(3,70,35,21,123,91,51,98)(4,73,44,32,124,94,60,109)(5,76,37,27,125,81,53,104)(6,79,46,22,126,84,62,99)(7,66,39,17,127,87,55,110)(8,69,48,28,128,90,64,105)(9,72,41,23,113,93,57,100)(10,75,34,18,114,96,50,111)(11,78,43,29,115,83,59,106)(12,65,36,24,116,86,52,101)(13,68,45,19,117,89,61,112)(14,71,38,30,118,92,54,107)(15,74,47,25,119,95,63,102)(16,77,40,20,120,82,56,97)>;
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), (1,80,33,31,121,85,49,108)(2,67,42,26,122,88,58,103)(3,70,35,21,123,91,51,98)(4,73,44,32,124,94,60,109)(5,76,37,27,125,81,53,104)(6,79,46,22,126,84,62,99)(7,66,39,17,127,87,55,110)(8,69,48,28,128,90,64,105)(9,72,41,23,113,93,57,100)(10,75,34,18,114,96,50,111)(11,78,43,29,115,83,59,106)(12,65,36,24,116,86,52,101)(13,68,45,19,117,89,61,112)(14,71,38,30,118,92,54,107)(15,74,47,25,119,95,63,102)(16,77,40,20,120,82,56,97) );
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)], [(1,80,33,31,121,85,49,108),(2,67,42,26,122,88,58,103),(3,70,35,21,123,91,51,98),(4,73,44,32,124,94,60,109),(5,76,37,27,125,81,53,104),(6,79,46,22,126,84,62,99),(7,66,39,17,127,87,55,110),(8,69,48,28,128,90,64,105),(9,72,41,23,113,93,57,100),(10,75,34,18,114,96,50,111),(11,78,43,29,115,83,59,106),(12,65,36,24,116,86,52,101),(13,68,45,19,117,89,61,112),(14,71,38,30,118,92,54,107),(15,74,47,25,119,95,63,102),(16,77,40,20,120,82,56,97)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8N | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | SD16 | C8.C4 | C8.Q8 |
| kernel | C16⋊1C8 | C8⋊2C8 | C16⋊5C4 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of C16⋊1C8 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 16 | 2 | 7 |
| 0 | 0 | 4 | 14 | 11 | 4 |
| 0 | 0 | 4 | 14 | 11 | 13 |
| 0 | 0 | 10 | 7 | 10 | 7 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 12 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 15 | 9 |
| 0 | 0 | 5 | 3 | 5 | 3 |
| 0 | 0 | 14 | 12 | 14 | 3 |
| 0 | 0 | 0 | 2 | 15 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,2,4,4,10,0,0,16,14,14,7,0,0,2,11,11,10,0,0,7,4,13,7],[8,12,0,0,0,0,12,9,0,0,0,0,0,0,0,5,14,0,0,0,2,3,12,2,0,0,15,5,14,15,0,0,9,3,3,0] >;
C16⋊1C8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_1C_8 % in TeX
G:=Group("C16:1C8"); // GroupNames label
G:=SmallGroup(128,100);
// by ID
G=gap.SmallGroup(128,100);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,36,422,100,1018,136,2804,172]);
// Polycyclic
G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^11>;
// generators/relations
Export