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