Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4466 |
| automorphism | Aut(G) = D3×#525 (order 384) · Inn(G) = C4×Dic3 · Out(G) = C23 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ #347 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×Dic3 |
| commutator | C3×C4 | C42 |
| Frattini | C2×C8 | C2×D3 |
| Fitting | C3×M32 | C2 |
| socle | C2×C3 | #97 |
| prime radical | C22×C3 | M16 |
| cosocle | C2×C3×C8 | C22 |
| hypercenter | M32 | D3 |
| hypocenter | C3 | #347 |
| LCS | G ▷ C3×C4 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ M32 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,11,7,9,13,16,8,10,12,17,14,15,19,18);(2,3)(5,9,10,15)(6,12)(11,18,17,16)(13,19)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=e,e^2=f,b^a=be,d^a=df,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#347, #69087, #69114, … |
| maximal normal subgroup of | D3×#347, #69087, #69114, … |
| maximal quotient of | #67511, #67512, #67811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 71);