Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C24 |
| direct product | C2×C11×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C11 | C22 |
| commutator | C2 | C22×C8×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| socle | C23×C11 | C2×C4 |
| prime radical | C24×C11 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C11 ◁ G | |
202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25);(3,4)(5,6)(7,8,9,11,10,12,13,14);(4,6)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 812);