Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×D16 ⋊ C2 |
| Sylow | C7 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6400 |
| commutator, cosocle | C7×C16 | C22 |
| Frattini | C16 | C2×D7 |
| Fitting | C7×D16 | C2 |
| socle | C2×C7 | D16 |
| hypercenter | D16 | D7 |
| hypocenter | C7 | D32 |
| LCS | G ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,4,6,7,5,3)(9,14)(10,19)(11,20)(12,24)(15,16)(17,33)(18,29)(21,38)(22,36)(23,27)(25,37)(26,28)(30,39)(31,32)(34,35);(2,3)(4,5)(6,7)(8,9)(10,14)(11,23)(12,26)(13,18)(15,19)(16,20)(17,24)(21,33)(22,29)(25,39)(27,35)(28,34)(30,38)(31,36)(32,37)Polycyclic presentation:
2^6,7|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 | D7×D32, #11727131, #11727133, … |
| maximal normal subgroup of | D7×D32, #11727131, #11727133, … |
| maximal quotient of | #11725381, #11725383, #11725413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 76);