Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#123 ⋊ C2 |
| Sylow | C7 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#123 | C2 |
| prime radical | C2×#1159 | C2 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #536 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,4,6,7,5,3)(9,11)(12,15,16,23)(13,18,19,22)(14,21,20,26)(17,25,24,27);(8,9)(10,11)(14,20)(17,24)(21,26)(25,27);(12,14)(13,17)(15,21)(16,20)(18,25)(19,24)(22,27)(23,26);(2,3)(4,5)(6,7)(12,13,16,19)(14,17,20,24)(15,22,23,18)(21,27,26,25)Polycyclic presentation:
2^6,7|a^2=f,d^2=f,d^a=df,g^a=g^6,c^b=cf,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #11726899, #11737754, C2×#90339, … |
| maximal normal subgroup of | #11726899, #11737754, C2×#90339, … |
| maximal quotient of | #11731498, #11731501, #11731509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1101);