Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#121 | C2 |
| hypercenter | #121 | D3 |
| hypocenter | C3 | D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #121 | |
428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 11 points:
(1,2,3)(5,7)(8,9,10,11);(4,5)(6,7)(9,11);(5,7);(2,3)(9,11)Polycyclic presentation:
2^6,3|d^2=ef,d^a=def,g^a=g^2,c^b=ce,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #80022, #80024, #80060, … |
| maximal normal subgroup of | #80022, #80024, #80060, … |
| maximal quotient of | #73887, #73893, #73896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1168);