Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3713 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #625 |
| 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 | |
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:
(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,18,19)(20,21,22);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,19);(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(21,22)Polycyclic presentation:
2^5,3^2|c^2=e,d^2=e,b^a=bd,d^a=de,g^a=g^2,d^b=de,f^b=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10590063, #10590064, #10590389, … |
| maximal normal subgroup of | #10590063, #10590064, #10590389, … |
| maximal quotient of | #10590267, #10590270, #10590278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 470);