Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C2×#6499874 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#99 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22⋊C4 |
| commutator | C22 | C22×C3×C4 |
| Frattini, cosocle | C23 | C23×C3 |
| prime radical | C22×C3×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ G | |
210 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 210 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) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(3,4,5);(7,10,11,13)(8,12);(6,7)(8,10)(9,11)(12,13)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #67694, #67695, C2×#4471, … |
| maximal subgroup of | #67694, #67695, C3×#1390, … |
| maximal normal subgroup of | #67694, #67695, C3×#1390, … |
| maximal quotient of | C2×C3×#323, C2×C3×#324, C3×#1459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 842);