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 | #3802 |
| 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:
(1,3,11,14,5,12,4,9)(2,6,13,16,8,15,7,10)(17,18,19)(20,21,22);(1,2,5,8)(3,10,12,16)(4,13,11,7)(6,14,15,9)(18,19);(3,9)(4,11)(6,10)(7,13)(12,14)(15,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=de,d^2=e,c^a=cd,d^a=de,g^a=g^2,c^b=cd,d^b=de,f^b=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10590236, #10590244, #10590245, … |
| maximal normal subgroup of | #10590236, #10590244, #10590245, … |
| maximal quotient of | #10590651, #10590714, #10590715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 458);