Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C22×D3×C8 (order 192) · Inn(G) = C2×D3 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C32 ⋊ C2 |
| Sylow | C3 ⋊ M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×D3 |
| commutator, socle | C2×C3 | C2×C16 |
| Fitting | C3×C32 | C2 |
| prime radical | C2×D3 | C16 |
| cosocle | C3×C16 | C22 |
| hypercenter | C32 | D3 |
| hypocenter | C3 | M64 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C16 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,6,11,7,12,14,23,8,13,15,24,17,25,26,33,9,10,16,20,18,21,27,30,19,22,28,31,29,32,34,35);(2,3)(5,10)(11,20)(12,21)(13,22)(23,30)(24,31)(25,32)(33,35)Polycyclic presentation:
2^6,3|b^2=c,c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4447, #69366, D3×M64, … |
| maximal normal subgroup of | C2×#4447, #69366, D3×M64, … |
| maximal quotient of | #67589, Dic3⋊C32, (C2×D3)⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 6);