Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Fitting | C3×Q8 | C2×D3 |
| socle | C2×C3 | C2×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,3,12,4,5,11,10,9)(2,6,16,7,8,15,14,13)(17,18,19);(1,2,5,8)(3,13,10,15,9,16,11,7,12,6,4,14)(18,19)Polycyclic presentation:
2^2,3^2,2^3|a^2=g,e^2=g,f^2=g,d^a=d^2,c^b=c^2,e^b=f,f^b=e,e^c=fg,f^c=ef,f^e=fg24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64197, #10596435, #10596440, … |
| maximal normal subgroup of | C2×#64197, #10596435, #10596440, … |
| maximal quotient of | #10593191, #10593193, #10593197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 845);