direct product, cyclic, abelian, monomial
Aliases: C78, also denoted Z78, SmallGroup(78,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C78 |
| C1 — C78 |
| C1 — C78 |
Generators and relations for C78
G = < a | a78=1 >
Smallest permutation representation of C78
►Regular action on 78 points
Generators in S78
(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)
G:=sub<Sym(78)| (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)>;
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) );
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)]])
C78 is a maximal subgroup of
Dic39
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 78A | ··· | 78X |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 |
| kernel | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of C78 ►in GL1(𝔽79) generated by
| 3 |
G:=sub<GL(1,GF(79))| [3] >;
C78 in GAP, Magma, Sage, TeX
C_{78} % in TeX
G:=Group("C78"); // GroupNames label
G:=SmallGroup(78,6);
// by ID
G=gap.SmallGroup(78,6);
# by ID
G:=PCGroup([3,-2,-3,-13]);
// Polycyclic
G:=Group<a|a^78=1>;
// generators/relations
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