Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C22×C4×D32 (order 576) · Inn(G) = D32 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D32 |
| commutator, socle | C2×C32 | C2×C8 |
| Fitting, cosocle | C32×C8 | C22 |
| prime radical | C2×C3×D3 | C8 |
| hypocenter | C32 | M32 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C8 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(18,19)(20,21,22);(2,6)(7,12)(8,13)(14,16)(17,18,19)(21,22)Polycyclic presentation:
2^5,3^2|b^2=c,c^2=d,d^2=e,b^a=be,g^a=g^2,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#798, #10590226, #10590227, … |
| maximal normal subgroup of | D3×#798, #10590226, #10590227, … |
| maximal quotient of | #10589103, (C2×D3)⋊(C3⋊C16), Dic3⋊(C3⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 191);