Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = D3×D8 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×SD32 ⋊ C2 |
| Sylow | C3 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D8 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×SD32 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | SD32 | D3 |
| hypocenter | C3 | #509 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
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)(4,5,8,9)(6,13,15,19)(7,16,14,11)(10,18,17,12);(5,10)(6,12)(7,14)(9,17)(11,19)(13,16)(15,18);(2,3)(5,9)(10,17)(11,16)(13,19)Polycyclic presentation:
2^6,3|c^2=f,d^2=ef,e^2=f,c^a=cf,g^a=g^2,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4914, #82024, #82030, … |
| maximal normal subgroup of | C2×#4914, #82024, #82030, … |
| maximal quotient of | #69177, #69180, #69186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 473);