Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C22 |
| commutator | C2 | C2×C42×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C22×C7 | C22×C4 |
| prime radical | C23×C7 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22,23,24,25,26,27);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 813);