p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C16⋊4C8, C4.16SD32, C42.312D4, C8.15M4(2), C4.6(C4⋊C8), C8.15(C2×C8), (C2×C8).37Q8, C8⋊1C8.6C2, (C4×C16).13C2, (C2×C16).15C4, (C2×C4).161D8, (C2×C4).28Q16, C2.4(C8⋊1C8), C2.1(C16⋊4C4), C4.4(C8.C4), (C4×C8).387C22, C2.2(C8.4Q8), C22.17(C2.D8), (C2×C8).217(C2×C4), (C2×C4).102(C4⋊C4), SmallGroup(128,104)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊4C8
G = < a,b | a16=b8=1, bab-1=a7 >
Smallest permutation representation of C16⋊4C8
►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 88 35 17 61 118 72 97)(2 95 36 24 62 125 73 104)(3 86 37 31 63 116 74 111)(4 93 38 22 64 123 75 102)(5 84 39 29 49 114 76 109)(6 91 40 20 50 121 77 100)(7 82 41 27 51 128 78 107)(8 89 42 18 52 119 79 98)(9 96 43 25 53 126 80 105)(10 87 44 32 54 117 65 112)(11 94 45 23 55 124 66 103)(12 85 46 30 56 115 67 110)(13 92 47 21 57 122 68 101)(14 83 48 28 58 113 69 108)(15 90 33 19 59 120 70 99)(16 81 34 26 60 127 71 106)
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), (1,88,35,17,61,118,72,97)(2,95,36,24,62,125,73,104)(3,86,37,31,63,116,74,111)(4,93,38,22,64,123,75,102)(5,84,39,29,49,114,76,109)(6,91,40,20,50,121,77,100)(7,82,41,27,51,128,78,107)(8,89,42,18,52,119,79,98)(9,96,43,25,53,126,80,105)(10,87,44,32,54,117,65,112)(11,94,45,23,55,124,66,103)(12,85,46,30,56,115,67,110)(13,92,47,21,57,122,68,101)(14,83,48,28,58,113,69,108)(15,90,33,19,59,120,70,99)(16,81,34,26,60,127,71,106)>;
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), (1,88,35,17,61,118,72,97)(2,95,36,24,62,125,73,104)(3,86,37,31,63,116,74,111)(4,93,38,22,64,123,75,102)(5,84,39,29,49,114,76,109)(6,91,40,20,50,121,77,100)(7,82,41,27,51,128,78,107)(8,89,42,18,52,119,79,98)(9,96,43,25,53,126,80,105)(10,87,44,32,54,117,65,112)(11,94,45,23,55,124,66,103)(12,85,46,30,56,115,67,110)(13,92,47,21,57,122,68,101)(14,83,48,28,58,113,69,108)(15,90,33,19,59,120,70,99)(16,81,34,26,60,127,71,106) );
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)], [(1,88,35,17,61,118,72,97),(2,95,36,24,62,125,73,104),(3,86,37,31,63,116,74,111),(4,93,38,22,64,123,75,102),(5,84,39,29,49,114,76,109),(6,91,40,20,50,121,77,100),(7,82,41,27,51,128,78,107),(8,89,42,18,52,119,79,98),(9,96,43,25,53,126,80,105),(10,87,44,32,54,117,65,112),(11,94,45,23,55,124,66,103),(12,85,46,30,56,115,67,110),(13,92,47,21,57,122,68,101),(14,83,48,28,58,113,69,108),(15,90,33,19,59,120,70,99),(16,81,34,26,60,127,71,106)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8P | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 | SD32 | C8.4Q8 |
| kernel | C16⋊4C8 | C8⋊1C8 | C4×C16 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C16⋊4C8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 7 | 1 |
| 0 | 16 | 7 |
| 2 | 0 | 0 |
| 0 | 3 | 3 |
| 0 | 3 | 14 |
G:=sub<GL(3,GF(17))| [16,0,0,0,7,16,0,1,7],[2,0,0,0,3,3,0,3,14] >;
C16⋊4C8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_4C_8 % in TeX
G:=Group("C16:4C8"); // GroupNames label
G:=SmallGroup(128,104);
// by ID
G=gap.SmallGroup(128,104);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,596,422,436,136,2804,172]);
// Polycyclic
G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^7>;
// generators/relations
Export