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 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#125 | C2 |
| prime radical | C22×D3 | C23 |
| hypercenter | #125 | D3 |
| hypocenter | C3 | #533 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #125 | |
222 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)(4,6)(5,8)(7,10)(9,11)(12,14,15,19)(13,16,17,18);(6,10)(8,11)(12,13,15,17)(14,18,19,16);(4,5,7,9)(6,11,10,8);(2,3)(6,10)(8,11)Polycyclic presentation:
2^6,3|b^2=f,c^2=e,d^2=e,d^a=df,g^a=g^2,d^b=df,d^c=def42 conjugacy classes — view the full character table.
| maximal subgroup of | #68362, #68365, #68369, … |
| maximal normal subgroup of | #68362, #68365, #68369, … |
| maximal quotient of | #74148, #74150, #74162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1247);