Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#73081 |
| Fitting | C2×C7×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | #1020⋊C2 |
| commutator, cosocle | C2×C22⋊A4 | C22×C7 |
| Frattini | C23 | C2×C7×S4 |
| Fitting | C2×C7×#120 | D3 |
| socle | C23×C7 | C2×S4 |
| hypocenter | C22⋊A4 | C7×D4 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C7 | |
494 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,4)(6,12,10,9)(7,11)(13,14,15,16,17,18,19);(1,2)(3,4)(5,6)(7,10,11,9,8,12)154 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#73081 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#73081 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(6,12,10,9)(7,11)(13,14,15,16,17,18,19), (1,2)(3,4)(5,6)(7,10,11,9,8,12));