Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3774 |
| automorphism | Aut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C23 |
| Frattini | C2 | C2×GO+(4,𝔽2) |
| Fitting | C32×C4 | D4 |
| socle | C2×C32 | C2×D4 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4)(2,5,6)(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16);(3,4)(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(1,2)(3,6)(4,5)(7,8)(9,16)(10,14)(11,19)(12,15)(13,17)(18,21)(20,22)Polycyclic presentation:
2^5,3^2|c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,d^b=de,f^b=g,g^b=f,f^d=f^2,g^d=g^230 conjugacy classes — view the full character table.
| maximal subgroup of | #10593162, #10593297, #10593298, … |
| maximal normal subgroup of | #10593162, #10593297, #10593298, … |
| maximal quotient of | #10593349, #10593350, #10593351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 871);