Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#14679 (order 1536) · Inn(G) = C22×D3 · Out(G) = #379 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×C4×C8 ⋊ C2 |
| Sylow | C3 ⋊ #433 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D3 |
| commutator, socle | C22×C3 | C22×C4 |
| Fitting | C3×C4×C8 | C2 |
| prime radical | C22×D3 | C2×C4 |
| cosocle | C2×C3×C4 | C23 |
| hypercenter | C4×C8 | D3 |
| hypocenter | C3 | #433 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,14,15,20,16,21,22,26)(13,18,19,25,17,23,24,27);(4,6,7,10)(5,8,9,11)(13,17)(18,23)(19,24)(25,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,19)(16,17)(20,25)(21,23)(22,24)(26,27)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #78448, C2×#4690, #78455, … |
| maximal normal subgroup of | #78448, C2×#4690, #78455, … |
| maximal quotient of | #68606, #68608, #68656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 249);