Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#497 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C7 |
| prime radical | C2×C7×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,4)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,23,24,25,26,27);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(1,2,3,4)(7,14)(10,19)(13,20)(15,18)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 905);