Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C23×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#126 ⋊ C2 |
| Sylow | C3 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#126 | C2 |
| prime radical | C2×#67 | C22 |
| hypercenter | #126 | D3 |
| hypocenter | C3 | #563 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #126 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,19)(14,21)(15,23)(24,26);(5,8)(9,11)(12,15)(13,18)(14,20)(16,23)(17,24)(19,22)(21,27)(25,26);(4,5)(6,9)(7,8)(10,11)(12,14,16,21)(13,17,19,25)(15,20,23,27)(18,24,22,26);(2,3)(6,10)(9,11)(12,13)(14,17)(15,22)(16,19)(18,23)(20,26)(21,25)(24,27)Polycyclic presentation:
2^6,3|b^2=f,d^2=e,c^a=cf,d^a=def,g^a=g^2,c^b=ce,d^b=df,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5712, #86110, #86118, … |
| maximal normal subgroup of | C2×#5712, #86110, #86118, … |
| maximal quotient of | #74265, #74280, #74281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1271);