Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#127 ⋊ C2 |
| Sylow | C3 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#127 | C2 |
| hypercenter | #127 | D3 |
| hypocenter | C3 | D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #127 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 11 points:
(1,2,3)(4,5,6,7)(8,10,11,9);(5,7)(8,9)(10,11);(5,7)(9,10);(2,3)(5,7)Polycyclic presentation:
2^6,3|d^2=f,d^a=def,g^a=g^2,c^b=ce,d^b=df,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68390, #68391, #68392, … |
| maximal normal subgroup of | #68390, #68391, #68392, … |
| maximal quotient of | #74326, #74335, #74360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1276);