Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×#11921467 (order 2304) · Inn(G) = C22⋊A4 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | C22×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C2×C3 |
| Frattini | C2 | C2×C22⋊A4 |
| Fitting | C22×#41 | C3 |
| socle | C23 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C4 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(5,6,9,12)(7,11,15,18,8,17,14,20,13,10,16,19);(5,13,9,15)(6,17,12,19)(7,16,14,8)(10,20,18,11);(1,4)(2,3)Polycyclic presentation:
2,3,2^5|a^2=g,e^2=g,f^2=g,c^b=d,d^b=cd,e^b=f,f^b=ef,f^e=fg24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1507);