Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C3 |
| socle | C2×C3 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 262 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)(3,10)(4,8)(5,7)(6,12)(9,14)(11,15)(13,16)(17,18,19);(1,3)(2,6)(4,11)(5,13)(7,15)(8,10)(9,16)(12,14);(2,7)(6,15)(8,9)(10,16);(1,4,5,14)(2,8,7,9)(3,11,13,12)(6,10,15,16)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #72112, #72113, #72114, … |
| maximal subgroup of | #72112, #72113, #72114, … |
| maximal normal subgroup of | #72112, #72113, #72114, … |
| maximal quotient of | C3×#2896, C3×#2911, C3×#2943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1464);