Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#124 | C2 |
| prime radical | C2×C3×D4 | C22 |
| hypercenter | #124 | D3 |
| hypocenter | C3 | #563 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #124 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,15,16,23)(13,18,19,26)(14,22,21,24)(17,25,20,27);(6,10)(8,11)(12,14,16,21)(13,17,19,20)(15,24,23,22)(18,27,26,25);(4,6)(5,8)(7,10)(9,11)(14,21)(15,23)(17,20)(18,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,19)(14,20,21,17)(15,18,23,26)(22,27,24,25)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,d^2=f,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 | #68354, #68355, #68359, … |
| maximal normal subgroup of | #68354, #68355, #68359, … |
| maximal quotient of | #74161, #74205, #74209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1231);