Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3714 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = #626 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #626 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
144 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,18,19)(20,21,22);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(18,19);(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=d,c^2=e,d^2=e,b^a=bd,d^a=de,g^a=g^2,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590341, #10590342, #10590923, … |
| maximal normal subgroup of | #10590341, #10590342, #10590923, … |
| maximal quotient of | #10590271, #10590274, #10590292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 477);