Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×C4×S4 (order 1152) · Inn(G) = C22 · Out(G) = C22×C3×C4×D3 |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | C22 |
| commutator | C2 | C22×C7×C8 |
| Frattini, cosocle | C8 | C23×C7 |
| socle | C2×C7 | C22×C8 |
| prime radical | C7×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ 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:
(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32)(33,34,35,36,37,38,39);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 912);