Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#123 | C2 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #563 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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)(9,13)(12,15)(18,25)(21,30)(24,27)(29,31);(10,14)(12,15)(16,19,20,28)(17,22,23,26)(18,24,25,27)(21,29,30,31);(8,10)(9,12)(11,14)(13,15)(16,18)(17,21)(19,27)(20,25)(22,31)(23,30)(24,28)(26,29);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,20,23)(18,21,25,30)(19,26,28,22)(24,31,27,29)Polycyclic presentation:
2^6,7|a^2=f,c^2=f,c^a=cf,d^a=def,g^a=g^6,c^b=ce,d^b=df61 conjugacy classes — view the full character table.
| maximal subgroup of | #11726895, #11728757, C2×#90342, … |
| maximal normal subgroup of | #11726895, #11728757, C2×#90342, … |
| maximal quotient of | #11731507, #11731540, #11731541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1104);