Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| 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 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,4,5,3,6,10,13,11)(2,7,8,12,9,15,14,16)(17,18,19)(20,21,22);(1,3,6,11)(2,7,9,15)(4,13,10,5)(8,16,14,12)(18,19);(1,2)(3,7)(4,12)(5,14)(6,9)(8,13)(10,16)(11,15)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,g^a=g^2,c^b=cd,d^b=de,f^b=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10590018, #10590019, #10593275, … |
| maximal normal subgroup of | #10590018, #10590019, #10593275, … |
| maximal quotient of | #10590652, #10590712, #10590713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 457);