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) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C22 |
| Fitting | C22×Q8 | D3 |
| prime radical | C22×SL(2,𝔽3) | C2 |
| hypercenter | C23 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,7,14)(6,10,18)(9,15,16)(12,19,20);(1,2)(3,4)(5,6,9,12)(7,11,15,17)(8,10,13,19)(14,20,16,18)Polycyclic presentation:
2^3,3,2^3|a^2=g,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 | #85319, #85322, #85329, … |
| maximal normal subgroup of | #85319, #85322, #85329, … |
| maximal quotient of | #73061, #73062, #73085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 978);