Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic36 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C9 ⋊ C2 |
| Sylow | C9 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D36 |
| commutator, cosocle | C4×C9 | C23 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C2×C8×C9 | C2 |
| socle | C2×C3 | C2×D12 |
| hypercenter | C2×C8 | D9 |
| hypocenter | C9 | #135 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,5,7,9,8,3,6,4)(10,13,14,21,15,22,12,19)(11,17,16,20,18,24,23,25);(10,12,15,14)(11,16,18,23)(13,19,22,21)(17,20,24,25);(2,4)(3,7)(5,6)(8,9)(10,11)(12,16)(13,20)(14,23)(15,18)(17,21)(19,24)(22,25)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=d,d^2=e,f^3=g^2,c^a=cd,d^a=de,f^a=f^2g,g^a=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10588165, #10588173, #10588187, … |
| maximal normal subgroup of | #10588165, #10588173, #10588187, … |
| maximal quotient of | #10588363, C4×Dic36, C4×#3664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 115);