Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C23×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#126 ⋊ C2 |
| Sylow | C3 ⋊ #552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#126 | C2 |
| prime radical | C2×#67 | C22 |
| hypercenter | #126 | D3 |
| hypocenter | C3 | #552 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #126 | |
214 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(12,14)(13,16)(15,18)(17,19);(5,8)(9,10)(12,13,15,17)(14,16,18,19);(4,5,7,8)(6,10,11,9)(14,18)(16,19);(2,3)(14,18)(16,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=ef,d^2=e,d^a=def,g^a=g^2,c^b=ce,d^b=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #86110, C2×#5708, #86116, … |
| maximal normal subgroup of | #86110, C2×#5708, #86116, … |
| maximal quotient of | #74264, #74266, #74268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1267);