Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×D3×D21 |
| Fitting, socle | C23×C32×C7 ⋊ C22 |
| derived, Hall | C32×C7 ⋊ C25 |
| Sylow | C32 ⋊ C24×D7 |
| Sylow | C7 ⋊ C23×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×D21 |
| commutator, cosocle, hypocenter | C32×C7 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C7 | C22 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C23 | |
3126 conjugacy classes of subgroups; only the 690 normal subgroups are shown.
This group has 3126 conjugacy classes of subgroups — too many to draw. Only its 690 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 5 generators on 19 points:
(5,6)(7,8,9)(10,11,12)(13,14,16,18,19,17,15);(3,4);(1,2)(3,4)(5,6)(8,9)(11,12)(14,15)(16,17)(18,19);(1,2);(11,12)288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×D32×D7, #72247046, #72247074 |
| maximal normal subgroup of | C23×D32×D7, #72247046, #72247074 |
| maximal quotient of | #72246737 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,9)(10,11,12)(13,14,16,18,19,17,15), (3,4), (1,2)(3,4)(5,6)(8,9)(11,12)(14,15)(16,17)(18,19), (1,2), (11,12));