direct product, cyclic, abelian, monomial
Aliases: C99, also denoted Z99, SmallGroup(99,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C99 |
| C1 — C99 |
| C1 — C99 |
Generators and relations for C99
G = < a | a99=1 >
Smallest permutation representation of C99
►Regular action on 99 points
Generators in S99
(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)
G:=sub<Sym(99)| (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)>;
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) );
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)]])
C99 is a maximal subgroup of
D99
99 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 11A | ··· | 11J | 33A | ··· | 33T | 99A | ··· | 99BH |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 11 | ··· | 11 | 33 | ··· | 33 | 99 | ··· | 99 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C9 | C11 | C33 | C99 |
| kernel | C99 | C33 | C11 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 10 | 20 | 60 |
Matrix representation of C99 ►in GL2(𝔽199) generated by
| 28 | 0 |
| 0 | 178 |
G:=sub<GL(2,GF(199))| [28,0,0,178] >;
C99 in GAP, Magma, Sage, TeX
C_{99} % in TeX
G:=Group("C99"); // GroupNames label
G:=SmallGroup(99,1);
// by ID
G=gap.SmallGroup(99,1);
# by ID
G:=PCGroup([3,-3,-11,-3,99]);
// Polycyclic
G:=Group<a|a^99=1>;
// generators/relations
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