Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C22×D3×D4 (order 192) · Inn(G) = #67 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #67 |
| commutator, cosocle | C3×C4 | C23 |
| Frattini | C4 | C22×D3 |
| Fitting | C3×#41 | C2 |
| socle | C2×C3 | C2×D4 |
| hypercenter | #41 | D3 |
| hypocenter | C3 | #135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,10)(7,15)(9,16)(11,12)(13,19)(17,18);(4,6,8,14)(5,9,12,18)(7,13,15,19)(10,16,11,17);(2,3)(4,5)(6,9)(7,10)(8,12)(11,15)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,3|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^2,d^c=de24 conjugacy classes — view the full character table.
| maximal subgroup of | #4823, #4824, #5144, … |
| maximal normal subgroup of | #4823, #4824, #5144, … |
| maximal quotient of | #4996, #4998, #5002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 157);