Finite Group DatabasebetaFinite Group Databasebeta| order | 22464 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| direct product | C4×SL(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | SL(3,𝔽3) |
| commutator, hypocenter | SL(3,𝔽3) | C4 |
| Frattini, cosocle | C2 | C2×SL(3,𝔽3) |
| socle, prime radical | C2×SL(3,𝔽3) | C2 |
| LCS | G ▷ SL(3,𝔽3) | |
| UCS | C1 ◁ C4 | |
263 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 6 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) = 52 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,3,2,4)(5,16,11,14)(6,10,13,7,12,9,17,8);(5,15,9,12)(7,17,14,16)(8,11)(10,13)48 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,4)(5,16,11,14)(6,10,13,7,12,9,17,8), (5,15,9,12)(7,17,14,16)(8,11)(10,13));