Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = C22×D3 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D3 |
| commutator, socle | C2×C3 | C22×C4 |
| Fitting, prime radical | C3×#41 | C2 |
| cosocle | C3×C4 | C23 |
| hypercenter | #41 | D3 |
| hypocenter | C3 | #131 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ #41 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(6,12)(9,15)(13,18)(16,19);(4,6)(5,9)(7,13)(8,12)(10,16)(11,15)(14,18)(17,19);(2,3)(4,5,7,10,8,11,14,17)(6,9,13,16,12,15,18,19)Polycyclic presentation:
2^5,3|a^2=d,d^2=e,f^a=f^2,c^b=ce30 conjugacy classes — view the full character table.
| maximal subgroup of | #4823, #4824, #5196, … |
| maximal normal subgroup of | #4823, #4824, #5196, … |
| maximal quotient of | #4997, #4999, #5000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 155);