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.
| Fitting | C7×Q32 ⋊ C2 |
| Sylow | C7 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6400 |
| commutator, cosocle | C7×C16 | C22 |
| Frattini | C16 | C2×D7 |
| Fitting | C7×Q32 | C2 |
| socle | C2×C7 | D16 |
| prime radical | D112 | C2 |
| hypercenter | Q32 | D7 |
| hypocenter | C7 | SD64 |
| LCS | G ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
32 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)(8,9,13,17)(10,19,22,36)(11,23,25,39)(12,26,24,16)(14,27,29,18)(15,31,32,20)(21,38,34,30)(28,35,33,37);(2,3)(4,5)(6,7)(9,14)(10,18)(11,20)(12,24)(15,30)(16,33)(17,29)(19,23)(21,37)(22,27)(25,31)(26,28)(32,38)(34,35)(36,39)Polycyclic presentation:
2^6,7|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 | D7×SD64, #11727134, #11727137, … |
| maximal normal subgroup of | D7×SD64, #11727134, #11727137, … |
| maximal quotient of | #11725382, #11725383, #11725632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 78);