Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×#72253807 |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C22 |
| derived | C9×H(3,𝔽3) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72253807 |
| commutator, cosocle, hypocenter | C9×H(3,𝔽3) | C22×C32 |
| Frattini | C32 | C32×D3×C3⋊D3 |
| Fitting | C3×H(3,𝔽3)×3−1+2 | C22 |
| socle | C33 | C3×D3×C3⋊D3 |
| LCS | G ▷ C9×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
4316 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 4316 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(5,7)(6,10)(8,11)(9,12)(13,17,16,15,14,19)(18,21,20);(4,8,9,6,12,11,10,5,7)(14,16)(15,21,17,18,19,20);(15,19,17)(18,21,20)219 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((1,2,3)(5,7)(6,10)(8,11)(9,12)(13,17,16,15,14,19)(18,21,20), (4,8,9,6,12,11,10,5,7)(14,16)(15,21,17,18,19,20), (15,19,17)(18,21,20));