Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of C3⋊#226 |
| automorphism | Aut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C3×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Frattini | C2 | C2×C3⋊S4 |
| Fitting | C3×#41 | D3 |
| socle | C2×C3 | C2×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,6)(2,11)(3,8)(4,16)(5,15)(7,10)(9,14)(12,13)(17,18,19);(3,4)(6,7)(9,11)(10,12)(13,15)(14,16)(18,19);(3,9,12)(4,10,11)(6,13,16)(7,14,15)Polycyclic presentation:
2^2,3^2,2^3|b^2=g,e^2=g,f^2=g,c^a=c^2,d^a=d^2,e^a=f,f^a=e,e^c=fg,f^c=ef,f^e=fg30 conjugacy classes — view the full character table.
| maximal subgroup of | #10593220, #10593222, #10593638, … |
| maximal normal subgroup of | #10593220, #10593222, #10593638, … |
| maximal quotient of | C4×C3⋊#226, #10593623, C4×C3⋊GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 914);