Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×C22⋊C4 ⋊ C2 |
| Sylow | C9 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D9 |
| commutator, cosocle | C22×C9 | C23 |
| Frattini, socle | C22×C3 | C22×D3 |
| Fitting | C9×C22⋊C4 | C2 |
| prime radical | C2×#611 | C2 |
| hypercenter | C22⋊C4 | D9 |
| hypocenter | C9 | #123 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
117 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,6,8,7,2,4,3,9,5)(11,13)(15,19)(18,21);(10,11)(12,13)(14,16,17,20)(15,18,19,21);(2,4)(3,7)(5,6)(8,9)(14,15)(16,18)(17,19)(20,21)Polycyclic presentation:
2^5,3^2|b^2=d,f^3=g^2,c^a=cd,f^a=f^2g,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63445, #10589592, #10589596, … |
| maximal normal subgroup of | C2×#63445, #10589592, #10589596, … |
| maximal quotient of | #10588332, #10588334, #10588337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 93);