Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C83 · Schur multiplier (H2) = C2×C4 |
| direct product | #431×C83 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C83 | C22 |
| commutator | C2 | C22×C8×C83 |
| Frattini, cosocle | C2×C4 | C23×C83 |
| socle | C22×C83 | C22×C4 |
| prime radical | C23×C83 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C83 ◁ G | |
118 conjugacy classes of subgroups.
3320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C83×#1274, C83×#1275, C83×#1276, … |
| maximal normal subgroup of | C83×#1274, C83×#1275, C83×#1276, … |
| maximal quotient of | C83×#1744, C4×C22⋊C8×C83, C4×C4⋊C8×C83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 112);