Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29×C3 |
| direct product | C23×C3⋊#1020 |
| Fitting, socle | C27×C3 ⋊ D3 |
| derived | C3×C22⋊A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×C22⋊A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C27×C3 | D3 |
| LCS | G ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C23 | |
24866 conjugacy classes of subgroups; only the 275 normal subgroups are shown.
This group has 24866 conjugacy classes of subgroups — too many to draw. Only its 275 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 — 4 generators on 17 points:
(5,6)(10,11,13)(15,16,17);(3,4)(7,8,9)(14,17)(15,16);(1,2);(8,9)(11,12)(16,17)144 conjugacy classes — view the full character table.
| maximal subgroup of | C23×D3×#1020, C2×#72230852, C2×#72231672, … |
| maximal normal subgroup of | C23×D3×#1020, C2×#72228412, C2×#72229182 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(10,11,13)(15,16,17), (3,4)(7,8,9)(14,17)(15,16), (1,2), (8,9)(11,12)(16,17));