Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C22×D3×#392 (order 1536) · Inn(G) = C2×D3 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #322 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C2×D3 |
| commutator | C2×C3 | C4×C8 |
| Fitting | C3×C4×C8 | C2 |
| socle, prime radical | C22×C3 | C42 |
| cosocle | C3×C42 | C22 |
| hypercenter | C4×C8 | D3 |
| hypocenter | C3 | #322 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ C4×C8 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,8,7,9,10,11)(13,16)(17,19);(2,3)(12,13,14,17,15,16,18,19)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,c^2=f,d^2=e,b^a=bf,g^a=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | C8×ΓL(1,𝔽25), #68047, #68048, … |
| maximal normal subgroup of | C8×ΓL(1,𝔽25), #68047, #68048, … |
| maximal quotient of | #67442, #67499, #67512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 13);