Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #5386 |
| automorphism | Aut(G) = C24×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×S4 |
| commutator, prime radical, cosocle | C2×SL(2,𝔽3) | C22 |
| Fitting | C4×Q8 | D3 |
| hypercenter | C2×C4 | S4 |
| hypocenter | SL(2,𝔽3) | Q8 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,4,8)(2,5,6,7)(9,12,19)(10,21,23,14,15,24)(11,17,20)(16,22);(1,2,4,6)(3,7,8,5)(9,10,11,14)(12,16,17,22)(13,15,18,21)(19,24,20,23)Polycyclic presentation:
2^3,3,2^3|a^2=c,b^2=cg,e^2=g,f^2=g,b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5389, #85318, #85329, … |
| maximal normal subgroup of | C2×#5389, #85318, #85329, … |
| maximal quotient of | #73059, #73062, #73064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 948);