Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C3×#72213983 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #21690 |
| commutator | C4×C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C4×C8 | C24×C3 |
| prime radical | C3×#491469 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#7846 ◁ G | |
3038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,5)(3,7)(6,8)(10,14,16,22)(11,18)(12,20)(15,21,17,24)(19,23)(25,26,27);(2,6)(5,8)(10,15,16,17)(11,19)(14,21,22,24)(18,23);(1,2,3,5,4,6,7,8);(9,10,11,14,12,17,18,21,13,16,19,22,20,15,23,24)264 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5)(3,7)(6,8)(10,14,16,22)(11,18)(12,20)(15,21,17,24)(19,23)(25,26,27), (2,6)(5,8)(10,15,16,17)(11,19)(14,21,22,24)(18,23), (1,2,3,5,4,6,7,8), (9,10,11,14,12,17,18,21,13,16,19,22,20,15,23,24));