Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×Q8 | C22 |
| hypercenter | Q8 | D32 |
| hypocenter | C32 | 2−1+4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
311 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 311 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) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,5,6,3)(2,7,9,13)(4,10,12,11)(8,15,16,14)(17,18,19)(20,21,22);(1,4,6,12)(2,8,9,16)(3,10,5,11)(7,14,13,15);(1,3,6,5)(2,7,9,13)(4,11,12,10)(8,15,16,14)(18,19);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,d^2=e,d^a=de,g^a=g^2,c^b=ce,f^b=f^2,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | #10594749, #10594753, #10594754, … |
| maximal normal subgroup of | #10594749, #10594753, #10594754, … |
| maximal quotient of | #10594318, #10594319, #10594322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 963);