Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×C22⋊Q8 ⋊ C2 |
| Sylow | C7 ⋊ #542 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C22⋊Q8 | C2 |
| prime radical | #6447 | C2 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #542 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 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) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(9,13)(10,15)(18,22)(20,23);(8,10)(9,12)(11,15)(13,14)(16,18,19,22)(17,20,21,23);(10,15)(12,14)(16,17,19,21)(18,20,22,23);(2,3)(4,5)(6,7)(8,9)(10,14)(11,13)(12,15)(18,22)(20,23)Polycyclic presentation:
2^6,7|b^2=e,c^2=e,c^a=cf,d^a=def,g^a=g^6,c^b=cef,d^c=df64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90331, #11743055, #11743057, … |
| maximal normal subgroup of | C2×#90331, #11743055, #11743057, … |
| maximal quotient of | #11731315, #11731320, #11731340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1093);