Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#6342371 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C23 |
| commutator | C22 | C22×C3×C4 |
| Frattini, cosocle | C23 | C23×C3 |
| prime radical | C24×C3 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
234 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,14,15);(1,3,4,2)(6,8);(10,12)84 conjugacy classes — view the full character table.
| Fitting subgroup of | #67636, #69594, #69595, … |
| maximal subgroup of | #67636, C3×#1326, #69594, … |
| maximal normal subgroup of | #67636, C3×#1326, #69594, … |
| maximal quotient of | C3×C22⋊#95, C3×#1436, C3×#1439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 818);