Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4656 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C23 |
| Fitting | C3×#95 | C2 |
| hypercenter | #95 | D3 |
| hypocenter | C3 | #400 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #95 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,6,7,5)(9,13)(12,15)(18,23)(20,22);(4,5,7,6)(8,10,11,14)(9,12,13,15)(16,18)(17,20)(19,23)(21,22);(2,3)(5,6)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^6,3|a^2=f,b^2=d,c^2=ef,b^a=be,c^a=cd,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68298, #68299, #68330, … |
| maximal normal subgroup of | #68298, #68299, #68330, … |
| maximal quotient of | #68642, #68643, #68644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 218);