Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | D7×#5464 |
| Fitting, derived | C7×H(3,𝔽4) ⋊ C2×C3 |
| Hall | D7×H(3,𝔽4) ⋊ C3 |
| Sylow | C7 ⋊ C2×#5464 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×H(3,𝔽4) | C2×C3 |
| Frattini | C22 | D7×C22⋊A4 |
| socle | C22×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(3,𝔽4) | |
| UCS | C1 | |
576 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3)(4,5)(6,7)(9,13,16)(10,15,22)(11,19,12)(14,21,17)(18,23,20);(1,2,4,6,7,5,3)(9,23)(10,18)(13,22)(14,15)(16,17)(20,21);(8,21,19,13)(9,18,23,17)(10,14,16,15)(11,20,12,22)45 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5464, D7×#3532⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽7)×#5464, D7×#3532⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,7)(9,13,16)(10,15,22)(11,19,12)(14,21,17)(18,23,20), (1,2,4,6,7,5,3)(9,23)(10,18)(13,22)(14,15)(16,17)(20,21), (8,21,19,13)(9,18,23,17)(10,14,16,15)(11,20,12,22));