direct product, p-group, abelian, monomial
Aliases: C8×C16, SmallGroup(128,42)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C8×C16 |
| C1 — C8×C16 |
| C1 — C8×C16 |
Generators and relations for C8×C16
G = < a,b | a8=b16=1, ab=ba >
Smallest permutation representation of C8×C16
►Regular action on 128 points
Generators in S128
(1 40 63 65 109 96 25 117)(2 41 64 66 110 81 26 118)(3 42 49 67 111 82 27 119)(4 43 50 68 112 83 28 120)(5 44 51 69 97 84 29 121)(6 45 52 70 98 85 30 122)(7 46 53 71 99 86 31 123)(8 47 54 72 100 87 32 124)(9 48 55 73 101 88 17 125)(10 33 56 74 102 89 18 126)(11 34 57 75 103 90 19 127)(12 35 58 76 104 91 20 128)(13 36 59 77 105 92 21 113)(14 37 60 78 106 93 22 114)(15 38 61 79 107 94 23 115)(16 39 62 80 108 95 24 116)
(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)
G:=sub<Sym(128)| (1,40,63,65,109,96,25,117)(2,41,64,66,110,81,26,118)(3,42,49,67,111,82,27,119)(4,43,50,68,112,83,28,120)(5,44,51,69,97,84,29,121)(6,45,52,70,98,85,30,122)(7,46,53,71,99,86,31,123)(8,47,54,72,100,87,32,124)(9,48,55,73,101,88,17,125)(10,33,56,74,102,89,18,126)(11,34,57,75,103,90,19,127)(12,35,58,76,104,91,20,128)(13,36,59,77,105,92,21,113)(14,37,60,78,106,93,22,114)(15,38,61,79,107,94,23,115)(16,39,62,80,108,95,24,116), (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)>;
G:=Group( (1,40,63,65,109,96,25,117)(2,41,64,66,110,81,26,118)(3,42,49,67,111,82,27,119)(4,43,50,68,112,83,28,120)(5,44,51,69,97,84,29,121)(6,45,52,70,98,85,30,122)(7,46,53,71,99,86,31,123)(8,47,54,72,100,87,32,124)(9,48,55,73,101,88,17,125)(10,33,56,74,102,89,18,126)(11,34,57,75,103,90,19,127)(12,35,58,76,104,91,20,128)(13,36,59,77,105,92,21,113)(14,37,60,78,106,93,22,114)(15,38,61,79,107,94,23,115)(16,39,62,80,108,95,24,116), (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) );
G=PermutationGroup([[(1,40,63,65,109,96,25,117),(2,41,64,66,110,81,26,118),(3,42,49,67,111,82,27,119),(4,43,50,68,112,83,28,120),(5,44,51,69,97,84,29,121),(6,45,52,70,98,85,30,122),(7,46,53,71,99,86,31,123),(8,47,54,72,100,87,32,124),(9,48,55,73,101,88,17,125),(10,33,56,74,102,89,18,126),(11,34,57,75,103,90,19,127),(12,35,58,76,104,91,20,128),(13,36,59,77,105,92,21,113),(14,37,60,78,106,93,22,114),(15,38,61,79,107,94,23,115),(16,39,62,80,108,95,24,116)], [(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)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8AV | 16A | ··· | 16BL |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 |
| kernel | C8×C16 | C82 | C4×C16 | C4×C8 | C2×C16 | C16 | C2×C8 | C8 |
| # reps | 1 | 1 | 2 | 4 | 8 | 32 | 16 | 64 |
Matrix representation of C8×C16 ►in GL2(𝔽17) generated by
| 8 | 0 |
| 0 | 4 |
| 14 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(17))| [8,0,0,4],[14,0,0,3] >;
C8×C16 in GAP, Magma, Sage, TeX
C_8\times C_{16} % in TeX
G:=Group("C8xC16"); // GroupNames label
G:=SmallGroup(128,42);
// by ID
G=gap.SmallGroup(128,42);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,64,100,136,172]);
// Polycyclic
G:=Group<a,b|a^8=b^16=1,a*b=b*a>;
// generators/relations
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