Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3713 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #107 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #625 |
| commutator | C32×C4 | C2×C4 |
| Frattini, hypercenter | C2×C4 | D32 |
| Fitting, cosocle | C2×C32×C4 | C22 |
| socle, prime radical | C22×C32 | D4 |
| hypocenter | C32 | #107 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,3,4,2)(6,9)(7,12)(10,11)(14,15)(16,17,18);(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,14,15)(17,18)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=c,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 | #10590061, #10590062, #10590068, … |
| maximal normal subgroup of | #10590061, #10590062, #10590068, … |
| maximal quotient of | #10589100, #10589140, #10589142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 224);