Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#11757163⋊C2 (order 23040) · Inn(G) = C24×D3 · Out(G) = C2×S5 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D3 |
| commutator, socle, cosocle | C2×C3 | C25 |
| Fitting | C3×2−1+4 | C2 |
| hypercenter | 2−1+4 | D3 |
| hypocenter | C3 | #585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(4,7,8,14)(5,11,10,18)(6,12,13,15)(9,17,16,19);(5,10)(9,16)(11,18)(17,19);(4,6,8,13)(5,9,10,16)(7,15,14,12)(11,19,18,17);(4,5)(6,9)(7,11)(8,10)(12,17)(13,16)(14,18)(15,19);(2,3)(7,14)(11,18)(12,15)(17,19)Polycyclic presentation:
2^6,3|c^2=f,e^2=f,e^a=ef,g^a=g^2,d^b=df,e^c=ef51 conjugacy classes — view the full character table.
| maximal subgroup of | #80284, #80286, #80298, … |
| maximal normal subgroup of | #80284, #80286, #80298, … |
| maximal quotient of | #85872, #85873, #85874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1527);