Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#120 ⋊ C2 |
| Sylow | C7 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6500 |
| commutator, cosocle | C23×C7 | C23 |
| Frattini | C23 | C22×D7 |
| Fitting | C7×#120 | C2 |
| socle | C2×C7 | #120 |
| hypercenter | #120 | D7 |
| hypocenter | C7 | B(4,𝔽2) |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(12,15);(8,10)(9,12)(11,14)(13,15);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^6,7|c^a=cd,e^a=ef,g^a=g^6,c^b=ce,d^b=df58 conjugacy classes — view the full character table.
| maximal subgroup of | #11726884, #11726885, #11726886, … |
| maximal normal subgroup of | #11726884, #11726885, #11726886, … |
| maximal quotient of | #11727251, #11727253, #11727254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 566);