Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #638 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C22×C3⋊D3 |
| Fitting | C32×#41 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(20,21,22);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,18,19);(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(18,19)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=g^2,d^c=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10591662, #10591663, #10592522, … |
| maximal normal subgroup of | #10591662, #10591663, #10592522, … |
| maximal quotient of | #10592695, #10592697, #10592701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 807);