Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#124 | C2 |
| prime radical | C2×#237 | C2 |
| hypercenter | #124 | D3 |
| hypocenter | C3 | #536 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #124 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,5,6,7)(9,15)(11,19)(13,21)(16,18);(5,7)(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)(8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23)Polycyclic presentation:
2^6,3|b^2=e,c^2=e,d^2=ef,d^a=de,g^a=g^2,c^b=ce,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #69159, #69164, #80285, … |
| maximal normal subgroup of | #69159, #69164, #80285, … |
| maximal quotient of | #74152, #74165, #74167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1234);