Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 73728 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #519 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator | C2×C3 | C23×C4 |
| Fitting | C22×C3×Q8 | C2 |
| socle, cosocle | C22×C3 | C24 |
| prime radical | C23×C3 | C23 |
| hypercenter | C22×Q8 | D3 |
| hypocenter | C3 | #519 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,15)(13,21)(14,23)(20,22);(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,20,21,22);(8,10,12,17)(9,13,15,21)(11,18,19,16)(14,22,23,20);(2,3)(4,5,6,7)(8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=f,d^a=df,g^a=g^2,c^b=cf54 conjugacy classes — view the full character table.
| maximal subgroup of | #69672, #70101, #70102, … |
| maximal normal subgroup of | #69672, #70101, #70102, … |
| maximal quotient of | #82144, #82155, #82156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1371);