Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D4 |
| commutator | C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C2×C3×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,15);(6,9)(7,12)(10,11);(1,3,4,2)(6,10)(9,11)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #79751, #79752, #79753, … |
| maximal subgroup of | #79751, #79752, #79753, … |
| maximal normal subgroup of | #79751, #79752, #79753, … |
| maximal quotient of | C3×SD16⋊C8, C3×#1585, C3×#1588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 873);