Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#124 | C2 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | #124 | D7 |
| hypocenter | C7 | #563 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #124 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,4,6,7,5,3)(9,13)(12,15)(16,19,20,27)(17,22,23,30)(18,26,25,28)(21,29,24,31);(10,14)(12,15)(16,18,20,25)(17,21,23,24)(19,28,27,26)(22,31,30,29);(8,10)(9,12)(11,14)(13,15)(18,25)(19,27)(21,24)(22,30);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,20,23)(18,24,25,21)(19,22,27,30)(26,31,28,29)Polycyclic presentation:
2^6,7|a^2=f,c^2=f,d^2=f,c^a=cf,d^a=def,g^a=g^6,c^b=ce,d^b=df,d^c=df61 conjugacy classes — view the full character table.
| maximal subgroup of | #11726108, #11726109, #11726113, … |
| maximal normal subgroup of | #11726108, #11726109, #11726113, … |
| maximal quotient of | #11731606, #11731650, #11731654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1125);