Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#10596770⋊C2 (order 13824) · Inn(G) = C24×D3 · Out(G) = C2×GO+(4,𝔽2) |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D3 |
| commutator, socle, cosocle | C2×C3 | C25 |
| Fitting | C3×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D3 |
| hypocenter | C3 | #585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
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)(5,11)(7,14)(9,16)(12,18);(6,13)(9,16)(12,18)(15,19);(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,15,16,19);(4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,18)(17,19);(2,3)(4,5)(6,9)(7,10)(8,11)(12,15)(13,16)(14,17)(18,19)Polycyclic presentation:
2^6,3|c^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 | #80277, #80279, #80291, … |
| maximal normal subgroup of | #80277, #80279, #80291, … |
| maximal quotient of | #85856, #85857, #85858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1525);