Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = S4×#120 (order 768) · Inn(G) = S4 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4 |
| Frattini, socle | C22 | C2×S4 |
| Fitting | C2×#41 | D3 |
| prime radical | C2×#41⋊C3 | C2 |
| cosocle | C2×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,10,16,12,7,18,9,19,14,6,15,20)(8,11,13,17);(1,2,3,4)(7,8)(10,11)(13,15)(14,16)(17,19)(18,20)Polycyclic presentation:
2^3,3,2^3|a^2=c,b^2=g,e^2=g,f^2=g,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #73121, #73122, #73123, … |
| maximal normal subgroup of | #73121, #73122, #73123, … |
| maximal quotient of | C4×#859, #73064, #73068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 983);