Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63734 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3774 |
| commutator, cosocle | C3⋊(C3⋊C8) | C22 |
| Frattini, hypercenter | C4 | GO+(4,𝔽2) |
| Fitting | C32×C4 | D4 |
| socle | C2×C32 | D8 |
| hypocenter | C32 | D16 |
| LCS | G ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3)(2,5,6)(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20);(1,2)(3,6)(4,5)(7,8)(9,12)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22)Polycyclic presentation:
2^5,3^2|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,c^b=cd,d^b=de,f^b=g,g^b=f,f^c=f^2,g^c=g^221 conjugacy classes — view the full character table.
| maximal subgroup of | #10593275, #10593276, #10593277, … |
| maximal normal subgroup of | #10593275, #10593276, #10593277, … |
| maximal quotient of | #10590024, #10590025, #10590026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 382);