Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3688 |
| automorphism | Aut(G) = C22×((C22⋊C9)⋊C2)⋊C3 (order 864) · Inn(G) = (C22⋊C9)⋊C2 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C4 |
| Frattini | C3×C4 | S4 |
| Fitting, prime radical | C3×#41 | D3 |
| socle | C2×C3 | SL(2,ℤ/4) |
| cosocle | #41⋊C9 | C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C4 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,3,6,10)(2,8)(4,12,13,21)(5,16)(7,24,9,17)(11,14,23,15)(18,22,19,20)(26,28)(27,31)(29,30)(32,33);(1,4,12)(2,7,11)(3,9,17)(5,15,20)(6,14,23)(8,19,21)(10,18,22)(13,24,16)(25,33,28,31,30,29,27,26,32)Polycyclic presentation:
2^2,3^2,2^3|a^2=b,b^2=g,c^3=d^2,e^2=g,f^2=g,c^a=c^2d,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 | #10589506, #10589507, C2×#63422, … |
| maximal normal subgroup of | #10589506, #10589507, C2×#63422, … |
| maximal quotient of | (Q8⋊C9)⋊C8, C2×#63422, #11720109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 70);