Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = D3×#442 (order 384) · Inn(G) = #213 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #213 |
| commutator, cosocle | C3×C8 | C22 |
| Frattini | C8 | C2×D3 |
| Fitting, prime radical | C3×D8 | C2 |
| socle | C2×C3 | D8 |
| hypercenter | D8 | D3 |
| hypocenter | C3 | SD32 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(5,9)(6,13)(7,14)(10,11)(12,18)(15,19)(16,17);(2,3)(4,5,8,12)(6,9,15,18)(7,16,14,11)(10,19,17,13)Polycyclic presentation:
2^5,3|a^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,c^b=cd,d^b=de18 conjugacy classes — view the full character table.
| maximal subgroup of | #4911, #4912, D3×SD32, … |
| maximal normal subgroup of | #4911, #4912, D3×SD32, … |
| maximal quotient of | #4490, #4492, #4562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 34);