Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D32 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | #3828 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
144 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(3,9)(6,12)(10,15)(13,16)(17,18,19)(20,21,22);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(18,19);(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(21,22)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=d,d^2=e,g^a=g^2,c^b=ce,f^b=f^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10590341, #10590342, #10590917, … |
| maximal normal subgroup of | #10590341, #10590342, #10590917, … |
| maximal quotient of | #10590254, #10590255, #10590257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 465);