Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C23×C32 ⋊ C22 |
| Sylow | C32 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C23×C32 | C22 |
| hypercenter | C23 | D32 |
| hypocenter | C32 | H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C23 | |
359 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 14 points:
(2,6)(5,8)(9,10,11)(12,13,14);(3,7)(5,8);(1,3)(2,5)(4,7)(6,8)(10,11);(1,2)(3,5)(4,6)(7,8)(13,14)Polycyclic presentation:
2^5,3^2|d^a=de,g^a=g^2,c^b=ce,f^b=f^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10590407, #10590408, #10590409, … |
| maximal normal subgroup of | #10590407, #10590408, #10590409, … |
| maximal quotient of | #10594811, #10594812, #10594813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 978);