Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C22×D4×#120 (order 1024) · Inn(G) = C2×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D4 |
| commutator | C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
162 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,3,4)(6,9)(7,11)(10,12)(13,14,15);(5,6)(7,12)(8,10)(9,11);(6,10)(9,12)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #70297, #70298, #70299, … |
| maximal subgroup of | #70297, #70298, #70299, … |
| maximal normal subgroup of | #70297, #70298, #70299, … |
| maximal quotient of | C3×#1477, C3×#1483, C3×#1489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 850);