Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D32 |
| commutator, socle, prime radical, cosocle | C22×C32 | C23 |
| Fitting | C2×C32×C4 | C22 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #125 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
125 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 96 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6)(9,13)(11,14)(15,17,18,22)(16,19,20,21);(2,3)(7,9,10,13)(8,11,12,14);(5,6)(7,8,10,12)(9,11,13,14)(15,16,18,20)(17,21,22,19)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=d,c^2=de,c^a=cde,g^a=g^2,c^b=cd,f^b=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10593849, #10593851, #10593861, … |
| maximal normal subgroup of | #10593849, #10593851, #10593861, … |
| maximal quotient of | #10591207, #10591210, #10591225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 494);