Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×#11920947 |
| Fitting | C2×C3×#379 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #10593630 |
| commutator, cosocle | C2×C3×C22⋊A4 | C23 |
| Frattini | C23 | C22×C3⋊S4 |
| Fitting | C2×C3×#379 | D3 |
| socle | C24×C3 | C2×S4 |
| hypocenter | C3×C22⋊A4 | C2×D4 |
| LCS | G ▷ C2×C3×C22⋊A4 ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C22 | |
3306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 3306 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(4,5)(7,9)(11,16,15,14)(12,13);(1,2)(12,17,13)(14,15,16);(3,4,5)(6,7,8,9)(10,11)(12,14)(13,16)(15,17)84 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72229194, C2×D3×#72991, C2×#72228412 |
| maximal normal subgroup of | C2×#72229194, C2×D3×#72991, C2×#72228412 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5)(7,9)(11,16,15,14)(12,13), (1,2)(12,17,13)(14,15,16), (3,4,5)(6,7,8,9)(10,11)(12,14)(13,16)(15,17));