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 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,9,12,8,6,7,14,10)(2,4,15,11,3,13,16,5)(17,18,19);(1,2,6,3)(4,10,12,9,11,16,13,8,14,7,5,15)(18,19)Polycyclic presentation:
2^2,3^2,2^3|a^2=g,b^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 | #10596435, C2×#64198, #10596439, … |
| maximal normal subgroup of | #10596435, C2×#64198, #10596439, … |
| maximal quotient of | #10593191, Dic3×#226, #10593194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 846);