Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C7 |
| socle | C2×C7 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(3,12)(4,8)(5,19)(6,11)(7,14)(9,17)(10,18)(13,25)(15,31)(16,28)(20,27)(21,24)(22,29)(23,30)(26,32)(33,34,35,36,37,38,39);(1,3)(2,7)(4,13)(5,20)(6,16)(8,22)(9,25)(10,27)(11,23)(12,19)(14,17)(15,18)(21,29)(24,28)(26,31)(30,32);(1,4,6,18)(2,8,11,10)(3,13,16,15)(5,17,21,32)(7,22,23,27)(9,24,26,19)(12,25,28,31)(14,29,30,20)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 916);