Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C22×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C32×D4 ⋊ C22 |
| Sylow | C32 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×D4 | C22 |
| hypercenter | D4 | D32 |
| hypocenter | C32 | H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
359 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 24 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,3,4,7)(2,6,5,8)(9,10,11)(12,13,14);(2,5)(3,7);(1,2)(3,6)(4,5)(7,8)(10,11);(3,7)(6,8)(13,14)Polycyclic presentation:
2^5,3^2|d^2=e,d^a=de,g^a=g^2,c^b=ce,f^b=f^2,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | #10594732, #10594734, #10594735, … |
| maximal normal subgroup of | #10594732, #10594734, #10594735, … |
| maximal quotient of | #10594032, #10594034, #10594035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 962);