Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×M32 ⋊ C2 |
| Sylow | C7 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D7 |
| commutator, socle | C2×C7 | C22×C8 |
| Fitting | C7×M32 | C2 |
| prime radical | #1157 | C4 |
| cosocle | C7×C8 | C23 |
| hypercenter | M32 | D7 |
| hypocenter | C7 | #504 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C8 ◁ M32 | |
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:
(1,2,4,6,7,5,3)(8,10,11,19,12,20,21,32,13,18,22,30,23,31,33,38)(9,15,16,27,17,28,29,37,14,24,25,34,26,35,36,39);(9,14)(10,18)(16,25)(17,26)(19,30)(20,31)(29,36)(32,38);(2,3)(4,5)(6,7)(8,9)(10,15)(11,16)(12,17)(13,14)(18,24)(19,27)(20,28)(21,29)(22,25)(23,26)(30,34)(31,35)(32,37)(33,36)(38,39)Polycyclic presentation:
2^6,7|c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^6,c^b=cf100 conjugacy classes — view the full character table.
| maximal subgroup of | #11726876, #11726877, #11727014, … |
| maximal normal subgroup of | #11726876, #11726877, #11727014, … |
| maximal quotient of | #11726822, #11726826, #11726828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 441);