Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, prime radical, cosocle | C22×C3 | C24 |
| Fitting | C3×C4⋊Q8 | C2 |
| hypercenter | C4⋊Q8 | D3 |
| hypocenter | C3 | #557 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,6)(5,10,11,9)(12,15,16,19)(13,17,14,18);(6,7)(9,10)(12,14,16,13)(15,17,19,18);(4,6,8,7)(5,9,11,10);(2,3)(4,5,8,11)(6,9,7,10)(12,13,16,14)(15,18,19,17)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=ef,d^2=f,d^a=def,g^a=g^2,c^b=ce,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68456, #68458, #68460, … |
| maximal normal subgroup of | #68456, #68458, #68460, … |
| maximal quotient of | #74323, #74325, #74330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1280);