Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator | C2×C32 | C22×C4 |
| Fitting | C32×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C22×C32 | C23 |
| hypercenter | C4⋊C4 | C3⋊D3 |
| hypocenter | C32 | C4×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,3,4,2)(7,11)(9,12)(16,17,18);(5,7,8,11)(6,9,10,12)(13,14,15);(1,2,4,3)(5,6,8,10)(7,9,11,12)(14,15)(17,18)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=d,c^2=de,f^a=f^2,g^a=g^2,c^b=cd60 conjugacy classes — view the full character table.
| maximal subgroup of | #10590161, #10590431, #10590432, … |
| maximal normal subgroup of | #10590161, #10590431, #10590432, … |
| maximal quotient of | #10592345, #10592348, #10592351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 744);