Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×((C22⋊C9)⋊C2)⋊C3 (order 864) · Inn(G) = C2×(C22⋊C9)⋊C2 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22 |
| Frattini, socle | C2×C3 | C2×S4 |
| Fitting | C2×C3×Q8 | D3 |
| hypercenter | C22 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,3,10)(2,13,15,6,7,16)(5,11,12)(8,14)(17,25,20,23,22,21,19,18,24);(1,2)(3,8)(4,7)(5,6)(9,13)(10,16)(11,14)(12,15)(18,20)(19,23)(21,22)(24,25)Polycyclic presentation:
2^2,3^2,2^3|c^3=d^2,e^2=g,f^2=g,b^a=bg,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,e^c=fg,f^c=ef,f^e=fg27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63689, #10593098, #10593101, … |
| maximal normal subgroup of | C2×#63689, #10593098, #10593101, … |
| maximal quotient of | #10589490, #10589491, #10589496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 337);