Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#124 | C2 |
| hypercenter | #124 | D3 |
| hypocenter | C3 | H(3,𝔽4) |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #124 | |
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 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) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(13,17)(14,18);(5,8)(6,11)(12,14)(13,16)(15,18)(17,19);(6,11)(9,10)(12,13)(14,19)(15,17)(16,18);(2,3)(4,5)(6,10)(7,8)(9,11)(14,18)(16,19)Polycyclic presentation:
2^6,3|d^2=ef,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 | #68350, #68352, #68353, … |
| maximal normal subgroup of | #68350, #68352, #68353, … |
| maximal quotient of | #74204, #74215, #74218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1242);