Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#6036 |
| derived | C22×C52 ⋊ C2×C3×C4 |
| Hall | C22×#6036 ⋊ C3 |
| Sylow | C52 ⋊ C2×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52×A4 |
| commutator, hypocenter | C22×C52 | C2×C3×C4 |
| Fitting, socle, cosocle | C23×C52 | C22×C3 |
| prime radical | C2×A4×C5⋊D5 | C2 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 ◁ C2 | |
342 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 18 points:
(6,7,8)(10,11)(12,13)(14,15)(16,17);(1,2,3,4)(5,8)(6,7)(9,10,12,13,11)(15,16)(17,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#87886, A4×#87830, A4×#87832, … |
| maximal normal subgroup of | A4×#87886, A4×#87830, A4×#87832, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7,8)(10,11)(12,13)(14,15)(16,17), (1,2,3,4)(5,8)(6,7)(9,10,12,13,11)(15,16)(17,18));