Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C23×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#126 ⋊ C2 |
| Sylow | C3 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#126 | C2 |
| hypercenter | #126 | D3 |
| hypocenter | C3 | #538 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #126 | |
250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,15,16,14)(13,17,18,19);(5,8)(9,11)(12,14,16,15)(13,17,18,19);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,19)(16,18);(2,3)(6,10)(9,11)Polycyclic presentation:
2^6,3|c^2=f,d^2=ef,d^a=de,g^a=g^2,c^b=ce,d^b=df36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5704, #86116, #86118, … |
| maximal normal subgroup of | C2×#5704, #86116, #86118, … |
| maximal quotient of | #74262, #74269, #74279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1263);