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, prime radical, cosocle | C2×C3×A4 | C22 |
| Fitting, socle | C23×C3 | C2×D3 |
| hypocenter | C3×A4 | Q8 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,14,15);(2,3,4)(5,6,8,10)(7,11,12,9)(14,15)Polycyclic presentation:
2^3,3^2,2^2|a^2=c,b^2=c,b^a=bc,e^a=e^2,d^b=d^2,f^b=g,g^b=f,f^d=g,g^d=fg24 conjugacy classes — view the full character table.
| maximal subgroup of | Dic6×S4, (C3⋊S4)⋊Q8, D3×A4⋊Q8, … |
| maximal normal subgroup of | Dic6×S4, (C3⋊S4)⋊Q8, D3×A4⋊Q8, … |
| maximal quotient of | #10593196, #10593199, #10593202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 852);