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 ⋊ #556 |
| 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 | #556 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #126 | |
212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 212 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)(5,9)(8,11)(12,15,16,13)(14,18,19,17);(4,6,7,10)(5,8,9,11)(12,14,16,19)(13,17,15,18);(4,5)(6,8)(7,9)(10,11)(14,19)(17,18);(2,3)(12,13,16,15)(14,18,19,17)Polycyclic presentation:
2^6,3|a^2=f,c^2=e,d^2=f,c^a=cf,g^a=g^2,c^b=cf,d^b=def39 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5702, #86110, #86116, … |
| maximal normal subgroup of | C2×#5702, #86110, #86116, … |
| maximal quotient of | #74263, #74264, #74267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1261);