Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = S42 (order 576) · Inn(G) = C22×A4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C3 |
| Fitting | H(5,𝔽2) | C3 |
| hypercenter | Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,11,21,9,4,19,6,18,12,3,15,24)(2,16,20,5,8,23,10,14,17,7,13,22);(3,9)(4,14)(5,7)(8,18)(11,13)(12,24)(15,16)(17,23)(19,21)(20,22)Polycyclic presentation:
2^2,3,2^3|a^2=f,b^2=f,d^2=f,e^2=f,b^a=bf,d^c=e,e^c=de,e^d=ef19 conjugacy classes — view the full character table.
| derived subgroup of | #226⋊A4, GL(2,𝔽3)⋊A4, #859⋊A4, … |
| maximal subgroup of | #5428, #5429, #576⋊C3, … |
| maximal normal subgroup of | #5428, #5429, #576⋊C3, … |
| maximal quotient of | #520⋊C3, C4⋊(#41⋊C3), Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 201);