Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C7 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting, prime radical | C2×C7×#41 | C2 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | #536 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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)(14,21)(17,25)(20,27)(24,26);(12,15,16,23)(13,18,19,22)(14,20,21,27)(17,24,25,26);(12,14)(13,17)(15,20)(16,21)(18,24)(19,25)(22,26)(23,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,16,19)(14,17,21,25)(15,22,23,18)(20,26,27,24)Polycyclic presentation:
2^6,7|a^2=f,c^2=f,c^a=cf,d^a=de,g^a=g^6,d^b=df82 conjugacy classes — view the full character table.
| maximal subgroup of | #11727800, #11727812, #11727813, … |
| maximal normal subgroup of | #11727800, #11727812, #11727813, … |
| maximal quotient of | #11739675, #11739679, #11739680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1278);