Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of D224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C32 ⋊ C2 |
| Sylow | C7 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D112 |
| commutator, cosocle | C7×C16 | C22 |
| Frattini | C16 | C2×D7 |
| Fitting | C7×C32 | C2 |
| socle | C2×C7 | D16 |
| prime radical | D112 | C2 |
| hypercenter | C32 | D7 |
| hypocenter | C7 | SD64 |
| LCS | G ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
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)(8,9,15,27,24,32,36,33,12,17,30,39,23,38,10,14,13,18,19,35,11,16,21,28,25,34,37,31,20,26,22,29);(2,3)(4,5)(6,7)(9,14)(10,19)(11,23)(12,25)(15,22)(16,31)(17,33)(18,29)(20,24)(21,30)(26,35)(27,38)(28,34)(32,39)(36,37)Polycyclic presentation:
2^6,7|b^2=cd,c^2=de,d^2=e,e^2=f,b^a=bc,c^a=cdf,d^a=def,e^a=ef,g^a=g^6115 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89244, #11727123, #11727127, … |
| maximal normal subgroup of | C2×#89244, #11727123, #11727127, … |
| maximal quotient of | #11725398, #11725402, #11725403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 6);