Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = D3×D12 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D12 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C32×C8 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C8 | D32 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
134 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 40 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:
(1,3,4,9,5,10,11,16)(2,7,8,15,6,12,13,14)(17,18,19)(20,21,22);(2,6)(3,9)(4,11)(7,14)(10,16)(12,15)(18,19);(1,2)(3,7)(4,8)(5,6)(9,15)(10,12)(11,13)(14,16)(21,22)Polycyclic presentation:
2^5,3^2|c^2=d,d^2=e,b^a=be,g^a=g^2,c^b=cd,d^b=de,f^b=f^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#904, #10594677, #10594679, … |
| maximal normal subgroup of | D3×#904, #10594677, #10594679, … |
| maximal quotient of | #10590690, #10590704, #10590706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 454);