Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C2×#72216662 (order 4096) · Inn(G) = C24 · Out(G) = C2×#3484 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
270 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,23);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(2,4)(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #71863, #71864, #71865, … |
| maximal subgroup of | #71863, #71864, #71865, … |
| maximal normal subgroup of | #71863, #71864, #71865, … |
| maximal quotient of | C3×#2310, C3×#2311, C3×#2317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1425);