Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #5450 |
| automorphism | Aut(G) = #11921467 (order 1152) · Inn(G) = G · Out(G) = D3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | H(3,𝔽4) ⋊ C3 |
| derived | C24 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24 | C22×C3 |
| Frattini, socle | C22 | C22×A4 |
| Fitting | H(3,𝔽4) | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,3)(2,6)(4,11,12,10,5,16)(7,15,9,14,8,13);(1,9,5,2)(3,15,11,14)(4,7,12,8)(6,16,13,10)Polycyclic presentation:
2^2,3,2^4|d^a=dfg,e^a=ef,d^b=df,e^b=eg,d^c=e,e^c=de,f^c=g,g^c=fg17 conjugacy classes — view the full character table.
| maximal subgroup of | #4625⋊C2, C2×#5450, #5450⋊C3, … |
| maximal normal subgroup of | #4625⋊C2, C2×#5450, #5450⋊C3, … |
| maximal quotient of | #2375⋊C3, #2785⋊C3, #73203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1009);