Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | D3×#87830 |
| derived, Hall | C3×C52 ⋊ C2×C42 |
| Sylow | C52 ⋊ C42×D3 |
| Sylow | C3 ⋊ C2×#87830 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D5⋊AGL(1,𝔽5) |
| commutator, hypocenter | C3×C52 | C2×C42 |
| Fitting, socle | C2×C3×C52 | C22×C4 |
| prime radical | C2×D3×C5⋊D5 | C22 |
| cosocle | C2×C3×C5⋊D5 | C23 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
540 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(5,6,7,8)(9,10,11,12);(2,3)(4,5,8,6,7);(10,11)(12,13)(14,15,16,17)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11717375, #72224612, C2×D3×#87830, … |
| maximal normal subgroup of | D3×#11717375, C2×D3×#87830, D3×#11717300, … |
| maximal quotient of | #72224612, C2×D3×#87830, D3×#11716761, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6,7,8)(9,10,11,12), (2,3)(4,5,8,6,7), (10,11)(12,13)(14,15,16,17));