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) = C25 |
No aliases for this group.
| Fitting | C3×#123 ⋊ C2 |
| Sylow | C3 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#123 | C2 |
| hypercenter | #123 | D3 |
| hypocenter | C3 | #547 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #123 | |
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7)(8,11,12,10)(9,14,13,15);(4,5)(6,7)(8,10,12,11)(9,14,13,15);(8,9)(10,14)(11,15)(12,13);(2,3)(9,13)(14,15)Polycyclic presentation:
2^6,3|c^2=f,d^2=f,b^a=bf,g^a=g^2,d^b=df,d^c=de39 conjugacy classes — view the full character table.
| maximal subgroup of | #86003, C2×#5655, #86012, … |
| maximal normal subgroup of | #86003, C2×#5655, #86012, … |
| maximal quotient of | #74060, #74063, #74064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1214);