Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of C3⋊(#41⋊C3) |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = D3×A4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊(#41⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊(#41⋊C3) |
| unambiguous semidirect product | (C2×#239)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C22×C3 |
| Frattini | C2 | C2×D3×A4 |
| Fitting | C2×C3×Q8 | C2×C3 |
| socle | C22×C3 | C2×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C22 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(4,5)(6,7,10,13)(8,18,17,19,14,12,15,20,9,11,16,21);(1,2)(3,4,5)(6,16,10,14)(7,20,13,18)(8,9,15,17)(11,12,19,21)42 conjugacy classes — view the full character table.
| maximal subgroup of | #10593191, #10593194, #10593197, … |
| maximal normal subgroup of | #10593191, #10593194, #10593197, … |
| maximal quotient of | C4×C3⋊(#41⋊C3), C4⋊(C3⋊(#41⋊C3)), #5570⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 921);