p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C16⋊2C8, C8.20M4(2), (C4×C8).3C4, C4.18(C4×C8), C8.21(C2×C8), (C2×C16).3C4, C8⋊C8.8C2, C16⋊5C4.5C2, C2.3(C8⋊C8), (C2×C4).82C42, C2.1(C16⋊C4), C4.14(C8⋊C4), (C4×C8).302C22, C42.291(C2×C4), (C2×C4).53M4(2), C22.11(C8⋊C4), (C2×C8).236(C2×C4), SmallGroup(128,45)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊C8
G = < a,b | a16=b8=1, bab-1=a5 >
Smallest permutation representation of C16⋊C8
►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 25 66 126 111 91 44 57)(2 22 75 115 112 88 37 62)(3 19 68 120 97 85 46 51)(4 32 77 125 98 82 39 56)(5 29 70 114 99 95 48 61)(6 26 79 119 100 92 41 50)(7 23 72 124 101 89 34 55)(8 20 65 113 102 86 43 60)(9 17 74 118 103 83 36 49)(10 30 67 123 104 96 45 54)(11 27 76 128 105 93 38 59)(12 24 69 117 106 90 47 64)(13 21 78 122 107 87 40 53)(14 18 71 127 108 84 33 58)(15 31 80 116 109 81 42 63)(16 28 73 121 110 94 35 52)
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), (1,25,66,126,111,91,44,57)(2,22,75,115,112,88,37,62)(3,19,68,120,97,85,46,51)(4,32,77,125,98,82,39,56)(5,29,70,114,99,95,48,61)(6,26,79,119,100,92,41,50)(7,23,72,124,101,89,34,55)(8,20,65,113,102,86,43,60)(9,17,74,118,103,83,36,49)(10,30,67,123,104,96,45,54)(11,27,76,128,105,93,38,59)(12,24,69,117,106,90,47,64)(13,21,78,122,107,87,40,53)(14,18,71,127,108,84,33,58)(15,31,80,116,109,81,42,63)(16,28,73,121,110,94,35,52)>;
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), (1,25,66,126,111,91,44,57)(2,22,75,115,112,88,37,62)(3,19,68,120,97,85,46,51)(4,32,77,125,98,82,39,56)(5,29,70,114,99,95,48,61)(6,26,79,119,100,92,41,50)(7,23,72,124,101,89,34,55)(8,20,65,113,102,86,43,60)(9,17,74,118,103,83,36,49)(10,30,67,123,104,96,45,54)(11,27,76,128,105,93,38,59)(12,24,69,117,106,90,47,64)(13,21,78,122,107,87,40,53)(14,18,71,127,108,84,33,58)(15,31,80,116,109,81,42,63)(16,28,73,121,110,94,35,52) );
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)], [(1,25,66,126,111,91,44,57),(2,22,75,115,112,88,37,62),(3,19,68,120,97,85,46,51),(4,32,77,125,98,82,39,56),(5,29,70,114,99,95,48,61),(6,26,79,119,100,92,41,50),(7,23,72,124,101,89,34,55),(8,20,65,113,102,86,43,60),(9,17,74,118,103,83,36,49),(10,30,67,123,104,96,45,54),(11,27,76,128,105,93,38,59),(12,24,69,117,106,90,47,64),(13,21,78,122,107,87,40,53),(14,18,71,127,108,84,33,58),(15,31,80,116,109,81,42,63),(16,28,73,121,110,94,35,52)])
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 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | M4(2) | M4(2) | C16⋊C4 |
| kernel | C16⋊C8 | C8⋊C8 | C16⋊5C4 | C4×C8 | C2×C16 | C16 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 4 | 4 | 4 |
Matrix representation of C16⋊C8 ►in GL6(𝔽17)
| 3 | 10 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 15 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 |
| 0 | 0 | 14 | 7 | 15 | 16 |
| 0 | 0 | 15 | 7 | 9 | 13 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 14 | 8 | 0 | 0 |
| 0 | 0 | 15 | 13 | 6 | 15 |
| 0 | 0 | 5 | 8 | 9 | 11 |
G:=sub<GL(6,GF(17))| [3,14,0,0,0,0,10,14,0,0,0,0,0,0,6,0,14,15,0,0,15,0,7,7,0,0,2,4,15,9,0,0,0,16,16,13],[8,1,0,0,0,0,8,9,0,0,0,0,0,0,9,14,15,5,0,0,16,8,13,8,0,0,0,0,6,9,0,0,0,0,15,11] >;
C16⋊C8 in GAP, Magma, Sage, TeX
C_{16}\rtimes C_8 % in TeX
G:=Group("C16:C8"); // GroupNames label
G:=SmallGroup(128,45);
// by ID
G=gap.SmallGroup(128,45);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,253,64,723,100,2019,136,172]);
// Polycyclic
G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^5>;
// generators/relations
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