Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×SD32 ⋊ C2 |
| Sylow | C7 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7×D8 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×SD32 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | SD32 | D7 |
| hypocenter | C7 | #508 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
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)(8,10,13,21)(9,15,14,18)(11,23,25,38)(12,26,24,20)(16,33,27,39)(17,35,28,31)(19,36,37,22)(29,34,30,32);(9,14)(10,19)(11,22)(12,24)(15,29)(16,32)(18,30)(20,38)(21,37)(23,26)(25,36)(27,34)(31,33)(35,39);(2,3)(4,5)(6,7)(8,9)(10,18)(11,16)(12,17)(13,14)(15,21)(19,29)(20,35)(22,34)(23,39)(24,28)(25,27)(26,31)(30,37)(32,36)(33,38)Polycyclic presentation:
2^6,7|c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^6,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef55 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89688, #11739469, D7×#508, … |
| maximal normal subgroup of | C2×#89688, #11739469, D7×#508, … |
| maximal quotient of | #11726928, #11726935, #11726936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 450);