Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | D7×#5933 |
| Sylow | C7 ⋊ C2×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C22 |
| Fitting | C7×H(5,𝔽2) | C2×D3 |
| socle | C2×C7 | C2×#1020 |
| prime radical | D7×#997 | C2 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
550 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 23 points:
(2,3)(4,5)(6,7)(10,11,17)(13,14,19)(16,23,22)(18,21,20);(1,2,4,6,7,5,3)(8,19,18,23,12,22,10,14)(9,20,16,11,15,17,13,21)65 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5933 |
| maximal normal subgroup of | AGL(1,𝔽7)×#5933 |
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)(10,11,17)(13,14,19)(16,23,22)(18,21,20), (1,2,4,6,7,5,3)(8,19,18,23,12,22,10,14)(9,20,16,11,15,17,13,21));