Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = C22×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C23 |
| Fitting | C22×C3×C4 | C2 |
| prime radical | C23×C3 | C22 |
| hypercenter | C22×C4 | D3 |
| hypocenter | C3 | C22⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7);(8,10,11,15)(9,12,13,14);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)Polycyclic presentation:
2^5,3|a^2=e,b^2=e,b^a=be,c^a=cd,f^a=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | #4452, #4467, #4721, … |
| maximal normal subgroup of | #4452, #4467, #4721, … |
| maximal quotient of | #4928, #4929, #4933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 131);