Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C2×C32×D4 ⋊ C2 |
| Sylow | C32 ⋊ #121 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C23 |
| Fitting | C2×C32×D4 | C2 |
| hypercenter | C2×D4 | C3⋊D3 |
| hypocenter | C32 | #121 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
282 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 14 points:
(4,5,6)(8,10)(12,14);(1,2,3)(11,12)(13,14);(2,3)(5,6)(7,8)(9,10)(12,14)Polycyclic presentation:
2^5,3^2|b^a=be,c^a=cd,f^a=f^2,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | #10595116, #10595123, #10595127, … |
| maximal normal subgroup of | #10595116, #10595123, #10595127, … |
| maximal quotient of | #10592641, #10592644, #10592647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 796);