Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = #3802 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C32×C8 ⋊ C22 |
| Sylow | C32 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3802 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C32×C8 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C8 | D32 |
| hypocenter | C32 | AGL(1,ℤ/8) |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
146 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3,5,4,7,8,6)(9,10,11)(12,13,14);(2,6)(3,8)(5,7)(10,11);(2,5)(3,8)(6,7)(13,14)Polycyclic presentation:
2^5,3^2|c^2=d,d^2=e,c^a=cd,d^a=de,g^a=g^2,c^b=cde,d^b=de,f^b=f^233 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63798, #10594685, #10594686, … |
| maximal normal subgroup of | C2×#63798, #10594685, #10594686, … |
| maximal quotient of | #10590649, #10590651, #10590652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 446);