Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
| direct product | SL(2,𝔽3)×PSU(3,𝔽2)⋊C3 |
| Fitting | C32×Q8 ⋊ C3×SL(2,𝔽3) |
| derived | Q8×PSU(3,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | Q8×PSU(3,𝔽2) | C32 |
| Fitting | C32×Q8 | C3×SL(2,𝔽3) |
| socle | C2×C32 | A4×SL(2,𝔽3) |
| LCS | G ▷ Q8×PSU(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
288 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,3,7,9)(2,5,4,8)(11,12,16)(14,15,17);(2,3,7)(5,8,6)(10,14,13,16)(11,17,15,12)70 conjugacy classes — view the full character table.
| derived subgroup of | #72244547, GL(2,𝔽3)×AGL(2,𝔽3) |
| maximal subgroup of | SL(2,𝔽3)×AGL(2,𝔽3), GL(2,𝔽3)×PSU(3,𝔽2)⋊C3, #72244547 |
| maximal normal subgroup of | SL(2,𝔽3)×AGL(2,𝔽3), GL(2,𝔽3)×PSU(3,𝔽2)⋊C3, #72244547 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,7,9)(2,5,4,8)(11,12,16)(14,15,17), (2,3,7)(5,8,6)(10,14,13,16)(11,17,15,12));