Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C4×S4 (order 576) · Inn(G) = C22 · Out(G) = C2×C3×C4×D3 |
| homology | Abelianization (H1) = C22×C3×C13 · Schur multiplier (H2) = C1 |
| direct product | C3×Q8×C13 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C13 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C13 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C13 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 312 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7)(8,11,10,9)(12,13,14,15,16,17,18,19,20,21,22,23,24);(4,8,6,10)(5,9,7,11)195 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 44);