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×#126 | C2 |
| prime radical | C23×C7 | C23 |
| hypercenter | #126 | D7 |
| hypocenter | C7 | #541 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #126 | |
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 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)(10,15)(13,14)(16,19,20,28)(17,22,23,26)(18,25,24,27)(21,30,29,31);(8,11,12,9)(10,14,15,13)(18,24)(21,29)(25,27)(30,31);(8,10)(9,13)(11,14)(12,15)(16,18,20,24)(17,21,23,29)(19,27,28,25)(22,31,26,30);(2,3)(4,5)(6,7)(8,9,12,11)(10,13,15,14)(16,17,20,23)(18,21,24,29)(19,26,28,22)(25,31,27,30)Polycyclic presentation:
2^6,7|a^2=f,b^2=e,c^2=ef,d^2=e,d^a=de,g^a=g^6,c^b=ce,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90392, #11743178, #11743184, … |
| maximal normal subgroup of | C2×#90392, #11743178, #11743184, … |
| maximal quotient of | #11731705, #11731706, #11731712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1154);