direct product, cyclic, abelian, monomial
Aliases: C110, also denoted Z110, SmallGroup(110,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C110 |
| C1 — C110 |
| C1 — C110 |
Generators and relations for C110
G = < a | a110=1 >
Smallest permutation representation of C110
►Regular action on 110 points
Generators in S110
(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)
G:=sub<Sym(110)| (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)>;
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) );
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)]])
C110 is a maximal subgroup of
Dic55
110 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11J | 22A | ··· | 22J | 55A | ··· | 55AN | 110A | ··· | 110AN |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C11 | C22 | C55 | C110 |
| kernel | C110 | C55 | C22 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 4 | 4 | 10 | 10 | 40 | 40 |
Matrix representation of C110 ►in GL1(𝔽331) generated by
| 282 |
G:=sub<GL(1,GF(331))| [282] >;
C110 in GAP, Magma, Sage, TeX
C_{110} % in TeX
G:=Group("C110"); // GroupNames label
G:=SmallGroup(110,6);
// by ID
G=gap.SmallGroup(110,6);
# by ID
G:=PCGroup([3,-2,-5,-11]);
// Polycyclic
G:=Group<a|a^110=1>;
// generators/relations
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