Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D12 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C32×C8 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C8 | D32 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,10,15,5,12,4,11)(2,7,13,16,6,14,8,9)(17,18,19)(20,21,22);(1,2,5,6)(3,9,12,16)(4,13,10,8)(7,15,14,11)(18,19);(2,6)(7,14)(8,13)(9,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=de,d^2=e,b^a=be,g^a=g^2,c^b=cd,d^b=de,f^b=f^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10590233, #10590238, #10590242, … |
| maximal normal subgroup of | #10590233, #10590238, #10590242, … |
| maximal quotient of | #10590690, #10590705, #10590707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 456);