Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 9216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#127 ⋊ C2 |
| Sylow | C3 ⋊ #560 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#127 | C2 |
| hypercenter | #127 | D3 |
| hypocenter | C3 | #560 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #127 | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,9)(8,11)(12,14,15,19)(13,17,16,18);(6,10)(8,11)(13,16)(14,19);(4,6)(5,8)(7,10)(9,11)(14,19)(17,18);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,16)(17,19)Polycyclic presentation:
2^6,3|d^2=f,c^a=cf,d^a=def,g^a=g^2,c^b=ce,d^b=df,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #69147, #70985, C2×#5720, … |
| maximal normal subgroup of | #69147, #70985, C2×#5720, … |
| maximal quotient of | #74350, #74353, #74354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1279);