Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊(C22⋊C9) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C2 |
| Frattini | C3 | #1020 |
| Fitting, socle | C24×C3 | D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,4)(3,7)(5,6)(8,9)(10,11,13,12)(14,15,17,16);(1,9,4,7,6,5,3,2,8)(11,12,13)(15,17,16)Polycyclic presentation:
2,3^2,2^4|b^3=c^2,b^a=b^2c,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,d^b=e,e^b=de,f^b=fg,g^b=f18 conjugacy classes — view the full character table.
| maximal subgroup of | #64188⋊C2, C2×#64188, #11724023, … |
| maximal normal subgroup of | #64188⋊C2, C2×#64188, #11724023, … |
| maximal quotient of | #10593105, #10593106, #10593107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 836);