Finite Group DatabasebetaFinite Group Databasebeta| order | 3360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C4×C5×GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4×C5 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C4×C5 |
| Frattini, cosocle | C2 | C2×C5×GL(3,𝔽2) |
| socle, prime radical | C2×C5×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C4×C5 | |
132 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,6,7,8,9)(10,11,13)(12,14,15);(10,12,16,13,11,14,15)120 conjugacy classes — view the full character table.
| automorphism group of | C23×C25 |
| outer automorphism group of | C23×C25 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(5,6,7,8,9)(10,11,13)(12,14,15), (10,12,16,13,11,14,15));