Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = C2×D3×A4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊#994 |
| unambiguous semidirect product | #1009⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C22×C3 |
| Fitting | C3×#41 | C2×C3 |
| socle | C2×C3 | C22×A4 |
| hypercenter | C4 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,11,21,6,4,19,7,13,12,3,14,22)(2,16,24,5,9,23,10,15,17,8,18,20)(25,26,27);(3,6)(4,11)(5,8)(9,15)(12,20)(13,14)(16,18)(17,22)(19,24)(21,23)(26,27)Polycyclic presentation:
2^2,3,2^2,3,2|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bg,f^a=f^2,d^c=de,e^c=d,e^d=eg33 conjugacy classes — view the full character table.
| maximal subgroup of | #10593233, #10593234, #10593235, … |
| maximal normal subgroup of | #10593233, #10593234, #10593235, … |
| maximal quotient of | SL(2,𝔽3)×Dic6, #5574⋊C3, C4⋊(C3⋊(#41⋊C3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 924);