Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#61451 (order 1536) · Inn(G) = C2×#213 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#213 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×#135 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | #135 | D3 |
| hypocenter | C3 | #509 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5)(6,13)(7,16)(8,10)(9,12)(11,14)(15,18)(17,19);(5,10)(9,17)(11,16)(13,18);(2,3)(5,9)(6,12)(7,14)(10,17)(11,18)(13,16)(15,19)Polycyclic presentation:
2^6,3|d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^2,c^b=cf,d^c=de,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69242, #69243, #69249, … |
| maximal normal subgroup of | #69242, #69243, #69249, … |
| maximal quotient of | #70649, #70652, #70653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 752);