Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×D4×AGL(1,ℤ/8) (order 1536) · Inn(G) = #938 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #938 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C2×C3×D8 | C2 |
| socle | C2×C3 | #121 |
| hypercenter | C2×D8 | D3 |
| hypocenter | C3 | #472 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,11)(6,14)(7,8)(10,17)(12,13)(15,16);(4,6,8,15,9,16,7,14)(5,10,12,18,13,17,11,19);(2,3)(4,5)(6,10)(7,11)(8,12)(9,13)(14,19)(15,18)(16,17)Polycyclic presentation:
2^6,3|b^2=e,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69260, #69261, #71471, … |
| maximal normal subgroup of | #69260, #69261, #71471, … |
| maximal quotient of | #70195, #70196, #70197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 719);