Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×S4×AGL(1,ℤ/8) (order 4608) · Inn(G) = C22×D12 · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D12 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×#131 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | #131 | D3 |
| hypocenter | C3 | #576 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,13,18,8,14,7,12)(5,10,16,19,9,15,11,17);(5,9)(10,15)(11,16)(17,19);(4,5)(6,10)(7,11)(8,9)(12,17)(13,16)(14,15)(18,19);(2,3)(6,12)(7,13)(10,17)(11,16)(14,18)(15,19)Polycyclic presentation:
2^6,3|d^2=ef,e^2=f,d^a=de,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #69126, #69128, #71094, … |
| maximal normal subgroup of | #69126, #69128, #71094, … |
| maximal quotient of | #78546, #78560, #78561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1311);