Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/8) (order 768) · Inn(G) = C4⋊S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | C22×C8 | D3 |
| socle | C23 | D12 |
| prime radical | C2×A4 | D4 |
| hypercenter | C8 | S4 |
| hypocenter | A4 | Q16 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13);(2,3)(5,6,9,12)(7,13,16,20)(8,17,14,11)(10,18,19,15)Polycyclic presentation:
2^4,3,2^2|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=g,g^a=f,f^e=g,g^e=fg22 conjugacy classes — view the full character table.
| maximal subgroup of | #73036, #73037, #73134, … |
| maximal normal subgroup of | #73036, #73037, #73134, … |
| maximal quotient of | #73014, #73015, #73016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 957);