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) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator | C2×C3 | C22×C4 |
| Fitting | C3×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C22×C3 | C23 |
| hypercenter | C4⋊C4 | D3 |
| hypocenter | C3 | C4×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,6,7,5)(10,14)(12,15);(8,10,11,14)(9,12,13,15);(2,3)(4,5,7,6)(8,9,11,13)(10,12,14,15)Polycyclic presentation:
2^5,3|a^2=e,b^2=d,c^2=de,f^a=f^2,c^b=cd30 conjugacy classes — view the full character table.
| maximal subgroup of | #4757, #4758, #4762, … |
| maximal normal subgroup of | #4757, #4758, #4762, … |
| maximal quotient of | #4646, #4649, #4652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 94);