Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×C4×D4 (order 192) · Inn(G) = C1 · Out(G) = C2×C3×C4×D4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C3×C4×C13 |
| multiplicative | (ℤ/371)ˣ, (ℤ/395)ˣ, (ℤ/471)ˣ, (ℤ/477)ˣ, (ℤ/507)ˣ, (ℤ/628)ˣ, (ℤ/676)ˣ, (ℤ/742)ˣ, (ℤ/790)ˣ, (ℤ/942)ˣ, (ℤ/954)ˣ, (ℤ/1014)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C3×C13 |
| socle, prime radical | C22×C3×C13 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 312 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4)(7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22);(5,6)312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 42);