Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C22⋊Q8 | C2 |
| prime radical | C23×C7 | C23 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #541 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(10,14)(12,15)(17,23)(21,29)(22,31)(28,30);(8,10,11,14)(9,12,13,15)(16,19,20,27)(17,22,23,31)(18,24,25,26)(21,28,29,30);(16,18,20,25)(17,21,23,29)(19,26,27,24)(22,30,31,28);(2,3)(4,5)(6,7)(8,9,11,13)(10,12,14,15)(16,17)(18,21)(19,22)(20,23)(24,28)(25,29)(26,30)(27,31)Polycyclic presentation:
2^6,7|a^2=f,b^2=e,c^2=ef,d^a=de,g^a=g^6,c^b=ce,d^c=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #11737582, #11737587, #11737622, … |
| maximal normal subgroup of | #11737582, #11737587, #11737622, … |
| maximal quotient of | #11731313, #11731316, #11731337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1078);