Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6400 |
| commutator, cosocle | C7×C16 | C22 |
| Frattini | C16 | C2×D7 |
| Fitting | C7×Q32 | C2 |
| socle, prime radical | C2×C7 | D16 |
| hypercenter | Q32 | D7 |
| hypocenter | C7 | Q64 |
| LCS | G ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,4,6,7,5,3)(8,10,14,24)(9,15,19,40)(11,26,30,60)(12,32,34,66)(13,36,33,23)(16,42,44,69)(17,46,47,71)(18,48,35,39)(20,49,51,25)(21,41,37,28)(22,45,54,27)(29,64,62,59)(31,38,65,67)(43,70,56,53)(50,57,55,63)(52,61,58,68);(2,3)(4,5)(6,7)(8,9,14,19)(10,20,24,51)(11,25,30,49)(12,31,34,65)(13,35,33,18)(15,37,40,21)(16,41,44,28)(17,45,47,27)(22,53,54,70)(23,55,36,50)(26,46,60,71)(29,63,62,57)(32,42,66,69)(38,64,67,59)(39,52,48,58)(43,61,56,68)Polycyclic presentation:
2^6,7|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6,c^b=cd,d^b=de,e^b=ef52 conjugacy classes — view the full character table.
| maximal subgroup of | #11727133, #11727135, D7×Q64, … |
| maximal normal subgroup of | #11727133, #11727135, D7×Q64, … |
| maximal quotient of | #11725381, #11725384, #11725632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 79);