Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ Q82 |
| 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 | Q82 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
212 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)(4,6)(5,9)(7,11)(8,10)(12,14,15,19)(13,16,17,18);(4,6,7,11)(5,9,10,8)(12,13,15,17)(14,18,19,16);(4,6,7,11)(5,8,10,9);(2,3)(4,5,7,10)(6,9,11,8)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=e,d^2=ef,b^a=bf,g^a=g^2,d^b=df,d^c=de39 conjugacy classes — view the full character table.
| maximal subgroup of | #68504, #68505, #68507, … |
| maximal normal subgroup of | #68504, #68505, #68507, … |
| maximal quotient of | #74323, #74331, #74335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1281);