Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×#23900 (order 1536) · Inn(G) = C22×D3 · Out(G) = D42 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, prime radical, cosocle | C22×C3 | C23 |
| Fitting | C2×C3×Q8 | C2 |
| hypercenter | C2×Q8 | D3 |
| hypocenter | C3 | C4⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
68 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)(5,7)(8,10,11,15)(9,12,13,14);(8,9,11,13)(10,14,15,12);(2,3)(4,5,6,7)Polycyclic presentation:
2^5,3|a^2=e,b^2=d,c^2=d,c^a=ce,f^a=f^2,c^b=cd24 conjugacy classes — view the full character table.
| maximal subgroup of | #4478, #4752, #4792, … |
| maximal normal subgroup of | #4478, #4752, #4792, … |
| maximal quotient of | #4973, #4975, #4976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 151);