Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = C22×D3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×M32 ⋊ C2 |
| Sylow | C3 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D3 |
| commutator, socle | C2×C3 | C22×C8 |
| Fitting | C3×M32 | C2 |
| prime radical | #235 | C4 |
| cosocle | C3×C8 | C23 |
| hypercenter | M32 | D3 |
| hypocenter | C3 | #504 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ M32 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,6,7,15,8,16,17,28,9,14,18,26,19,27,29,34)(5,11,12,23,13,24,25,33,10,20,21,30,22,31,32,35);(5,10)(6,14)(12,21)(13,22)(15,26)(16,27)(25,32)(28,34);(2,3)(4,5)(6,11)(7,12)(8,13)(9,10)(14,20)(15,23)(16,24)(17,25)(18,21)(19,22)(26,30)(27,31)(28,33)(29,32)(34,35)Polycyclic presentation:
2^6,3|c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^2,c^b=cf60 conjugacy classes — view the full character table.
| maximal subgroup of | #69122, #69123, #69260, … |
| maximal normal subgroup of | #69122, #69123, #69260, … |
| maximal quotient of | #69068, #69072, #69074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 466);