Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4466 |
| automorphism | Aut(G) = D3×#2010 (order 768) · Inn(G) = C4×Dic3 · Out(G) = C2×D4 |
| 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×#97 | C2 |
| socle | C2×C3 | #97 |
| prime radical | C22×C3 | M16 |
| cosocle | C2×C3×C8 | C22 |
| hypercenter | #97 | D3 |
| hypocenter | C3 | #347 |
| LCS | G ▷ C3×C4 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ #97 | |
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)(5,9,11,12)(6,13)(10,18,19,17)(14,16);(2,3)(4,5,6,10,7,12,14,18,8,11,13,19,15,9,16,17)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,c^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 | #67520, #67521, #67783, … |
| maximal normal subgroup of | #67520, #67521, #67783, … |
| maximal quotient of | #67442, #67663, C2×#4466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 25);