Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C3×(C3⋊D3)⋊C32 |
| Fitting, Sylow | C3×H(3,𝔽3)2 ⋊ C22 |
| derived | C34 ⋊ C22×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C3⋊D3)⋊C32 |
| commutator, cosocle, hypocenter | C34 | C22×C33 |
| Frattini | C32 | C33×D32 |
| Fitting | C3×H(3,𝔽3)2 | C22 |
| socle | C33 | C32×D32 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C3 | |
6778 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 6778 conjugacy classes of subgroups — too many to draw. Only its 318 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:
(5,7)(6,9)(8,10)(11,12)(13,14,17,18,16,19)(15,21);(4,5,11)(6,8,10)(7,9,12)(14,20,19,15,18,21)(16,17);(1,2,3)300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#72214399 |
| maximal normal subgroup of | C3×#72214399 |
| maximal quotient of | #72243756, #72243757, #72243840, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,7)(6,9)(8,10)(11,12)(13,14,17,18,16,19)(15,21), (4,5,11)(6,8,10)(7,9,12)(14,20,19,15,18,21)(16,17), (1,2,3));