Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | Aut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | GO+(4,𝔽2) |
| commutator | C3⋊D3 | C2×C8 |
| Fitting | C32×C4 | D4 |
| socle | C2×C32 | C4⋊C4 |
| prime radical | C2×C3⋊D3 | C2×C4 |
| cosocle | C4×C3⋊D3 | C22 |
| hypocenter | C32 | C4⋊C8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3)(2,5,6)(7,9,10,14,11,8,12,13);(1,2)(3,6)(4,5)(7,8,10,13,11,9,12,14)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=d,d^2=e,b^a=bc,f^a=f^2,f^b=g,g^b=f,f^c=f^2,g^c=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | PSU(3,𝔽2)⋊C8, #10593157, #10593158, … |
| maximal normal subgroup of | PSU(3,𝔽2)⋊C8, #10593157, #10593158, … |
| maximal quotient of | #10590022, #10590023, #10590041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 380);