direct product, cyclic, abelian, monomial
Aliases: C76, also denoted Z76, SmallGroup(76,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C76 |
| C1 — C76 |
| C1 — C76 |
Generators and relations for C76
G = < a | a76=1 >
Smallest permutation representation of C76
►Regular action on 76 points
Generators in S76
(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)
G:=sub<Sym(76)| (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)>;
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) );
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)]])
C76 is a maximal subgroup of
C19⋊C8 Dic38 D76
76 conjugacy classes
| class | 1 | 2 | 4A | 4B | 19A | ··· | 19R | 38A | ··· | 38R | 76A | ··· | 76AJ |
| order | 1 | 2 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C19 | C38 | C76 |
| kernel | C76 | C38 | C19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 |
Matrix representation of C76 ►in GL1(𝔽229) generated by
| 52 |
G:=sub<GL(1,GF(229))| [52] >;
C76 in GAP, Magma, Sage, TeX
C_{76} % in TeX
G:=Group("C76"); // GroupNames label
G:=SmallGroup(76,2);
// by ID
G=gap.SmallGroup(76,2);
# by ID
G:=PCGroup([3,-2,-19,-2,114]);
// Polycyclic
G:=Group<a|a^76=1>;
// generators/relations
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