Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#125 ⋊ C2 |
| Sylow | C3 ⋊ #562 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#125 | C2 |
| hypercenter | #125 | D3 |
| hypocenter | C3 | #562 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #125 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,6)(5,9,10,11)(12,14,15,18)(13,17,16,19);(4,6,8,7)(5,9,10,11)(12,13)(14,17)(15,16)(18,19);(4,5,8,10)(6,9,7,11)(13,16)(14,18);(2,3)(6,7)(9,11)(13,16)(17,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=e,d^2=f,c^a=cf,d^a=de,g^a=g^2,c^b=cef,d^b=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #68370, #68373, #68374, … |
| maximal normal subgroup of | #68370, #68373, #68374, … |
| maximal quotient of | #74198, #74201, #74219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1259);