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.
| Sylow | C3 ⋊ #552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#123 | C2 |
| prime radical | C2×C3×D4 | C22 |
| hypercenter | #123 | D3 |
| hypocenter | C3 | #552 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #123 | |
214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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)(6,11)(8,10)(13,17)(16,19);(4,6)(5,8)(7,11)(9,10)(12,14,15,18)(13,16,17,19);(12,13)(14,16)(15,17)(18,19);(2,3)(4,5,7,9)(6,10,11,8)Polycyclic presentation:
2^6,3|a^2=f,c^2=e,c^a=cf,g^a=g^2,d^b=de,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5650, #86001, #86003, … |
| maximal normal subgroup of | C2×#5650, #86001, #86003, … |
| maximal quotient of | #74059, #74060, #74067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1209);