Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C23×C3×C9 (order 216) · Inn(G) = C1 · Out(G) = C23×C3×C9 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C756 |
| abelian | C4×C7×C27 |
| multiplicative | (ℤ/757)ˣ, (ℤ/1514)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C9 | C2×C3×C7 |
| socle, prime radical | C2×C3×C7 | C2×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 756 |
A minimal faithful permutation representation — 1 generator on 38 points:
(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)756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 6);