Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×C22⋊Q8 ⋊ C2 |
| Sylow | C7 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #547 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
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)(12,14)(13,16)(15,18)(17,19);(9,11)(12,13,15,17)(14,16,18,19);(8,9,10,11)(14,18)(16,19);(2,3)(4,5)(6,7)(14,18)(16,19)Polycyclic presentation:
2^6,7|b^2=e,c^2=ef,d^a=def,g^a=g^6,c^b=ce,d^b=df67 conjugacy classes — view the full character table.
| maximal subgroup of | #11737602, #11737612, #11737650, … |
| maximal normal subgroup of | #11737602, #11737612, #11737650, … |
| maximal quotient of | #11731330, #11731338, #11731345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1084);