Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D32 |
| commutator | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| socle, prime radical, cosocle | C22×C32 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C4×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
155 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 18 points:
(5,7,8,12)(6,9,10,11)(13,14,15)(16,17,18);(1,2,3,4)(7,12)(9,11)(14,15);(5,6,8,10)(7,11,12,9)(17,18)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=d,c^2=e,c^a=ce,g^a=g^2,c^b=ce,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590482, #10590483, #10590484, … |
| maximal normal subgroup of | #10590482, #10590483, #10590484, … |
| maximal quotient of | #10590250, #10590252, #10590259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 491);