Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = C22×D4×S4 (order 768) · Inn(G) = C22×D4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 ◁ G | |
258 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 158 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:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(17,18,19);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #84012, #84013, #84014, … |
| maximal subgroup of | #84012, #84013, #84014, … |
| maximal normal subgroup of | #84012, #84013, #84014, … |
| maximal quotient of | C3×#2897, C3×#2898, C3×#2912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1466);