Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = C22×D3 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, prime radical, cosocle | C22×C3 | C23 |
| Fitting | C3×C4⋊C4 | C2 |
| hypercenter | C4⋊C4 | D3 |
| hypocenter | C3 | #125 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(6,10)(12,14,15,19)(13,16,17,18);(4,6,7,10)(5,8,9,11);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^5,3|a^2=d,b^2=e,c^2=d,c^a=cde,f^a=f^2,c^b=ce24 conjugacy classes — view the full character table.
| maximal subgroup of | #4763, #4764, #4766, … |
| maximal normal subgroup of | #4763, #4764, #4766, … |
| maximal quotient of | #4654, #4655, #4660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 97);