Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C8 · Schur multiplier (H2) = C26×C3 |
| direct product | C2×C32×#1580 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C2×D4 |
| commutator | C4 | C23×C32×C8 |
| Frattini, cosocle | C42 | C24×C32 |
| socle | C23×C32 | C4×D4 |
| prime radical | C22×C32×D8 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ C2×C32×C4×C8 ◁ G | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,4,6,8,7,9,10,5)(12,15)(13,17)(16,18)(19,20,21);(3,5,10,9,7,8,6,4)(12,16)(15,18)(22,23,24);(1,2);(11,12,13,15,14,16,17,18)792 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((3,4,6,8,7,9,10,5)(12,15)(13,17)(16,18)(19,20,21), (3,5,10,9,7,8,6,4)(12,16)(15,18)(22,23,24), (1,2), (11,12,13,15,14,16,17,18));