Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C22×D42 (order 256) · Inn(G) = #121 · Out(G) = C23 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G | |
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 19 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,19);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)48 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#471, #80929, #80930, … |
| maximal subgroup of | D3×#471, #80929, #80930, … |
| maximal normal subgroup of | D3×#471, #80929, #80930, … |
| maximal quotient of | C3×#1665, C3×#1666, C3×#1671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 904);