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 | C22 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C4 |
| Frattini, socle, prime radical | C22×C3 | S4 |
| Fitting | C2×C3×Q8 | D3 |
| cosocle | C2×Q8⋊C9 | C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4)(5,6,12,7,8,11,10,9)(14,16)(15,19)(17,18)(20,21);(6,9,12)(7,10,11)(13,21,16,19,18,17,15,14,20)Polycyclic presentation:
2^2,3^2,2^3|a^2=b,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 | #10589488, C4×#3688, #10589490, … |
| maximal normal subgroup of | #10589488, C4×#3688, #10589490, … |
| maximal quotient of | (Q8⋊C9)⋊C8, C2×#63421, #11720108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 69);