Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | GL(2,𝔽3)×((C3⋊D3)⋊C3)⋊C2 |
| Fitting | Q8×H(3,𝔽3) ⋊ C22×D3 |
| derived | SL(2,𝔽3)×H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | SL(2,𝔽3)×H(3,𝔽3) | C23 |
| Frattini, socle | C2×C3 | D32×S4 |
| Fitting | Q8×H(3,𝔽3) | C22×D3 |
| LCS | G ▷ SL(2,𝔽3)×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
1810 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 generators on 17 points:
(1,2,8,4,7,6)(3,5)(9,10,17,15,14,16)(11,13,12);(10,12)(13,17)(14,16);(2,5)(3,7)(6,8)88 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,8,4,7,6)(3,5)(9,10,17,15,14,16)(11,13,12), (10,12)(13,17)(14,16), (2,5)(3,7)(6,8));