Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C22×#6342371 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C23×C3 |
| prime radical | C22×C3×D4 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
238 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 78 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)(10,12)(13,14,15);(5,6)(7,8)(9,10)(11,12);(2,4)(6,8)(9,10,11,12)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #67647, #69702, #69703, … |
| maximal subgroup of | #67647, C3×#1345, #69702, … |
| maximal normal subgroup of | #67647, C3×#1345, #69702, … |
| maximal quotient of | C3×#1438, C3×#1439, C3×#1440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 827);