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 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#123 | C2 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #547 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(12,15,16,14)(13,18,17,19);(8,9)(10,11)(12,14,16,15)(13,18,17,19);(12,13)(14,18)(15,19)(16,17);(2,3)(4,5)(6,7)(13,17)(18,19)Polycyclic presentation:
2^6,7|c^2=f,d^2=f,b^a=bf,g^a=g^6,d^b=df,d^c=de67 conjugacy classes — view the full character table.
| maximal subgroup of | #11743071, C2×#90346, #11743080, … |
| maximal normal subgroup of | #11743071, C2×#90346, #11743080, … |
| maximal quotient of | #11731505, #11731508, #11731509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1108);