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 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #213 |
| commutator, cosocle | C3×C8 | C22 |
| Frattini | C8 | C2×D3 |
| Fitting | C3×Q16 | C2 |
| socle, prime radical | C2×C3 | D8 |
| hypercenter | Q16 | D3 |
| hypocenter | C3 | Q32 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,6,9,17)(5,10,13,27)(7,19,22,35)(8,24,20,16)(11,29,30,34)(12,31,23,26)(14,32,33,18)(15,28,25,21);(2,3)(4,5,9,13)(6,14,17,33)(7,18,22,32)(8,23,20,12)(10,25,27,15)(11,28,30,21)(16,34,24,29)(19,31,35,26)Polycyclic presentation:
2^5,3|a^2=e,b^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 | #4915, #4916, D3×Q32, … |
| maximal normal subgroup of | #4915, #4916, D3×Q32, … |
| maximal quotient of | #4489, #4492, #4564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 36);