Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×M16 ⋊ C2 |
| Sylow | C7 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D28 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×M16 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | C2×M16 | D7 |
| hypocenter | C7 | #575 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×M16 | |
262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,12,20,13,19,10,17)(9,15,14,18,16,22,21,23);(8,10,13,12)(9,14,16,21)(11,17,19,20)(15,18,22,23);(11,19)(15,22)(17,20)(18,23);(2,3)(4,5)(6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,20)(17,22)(19,23)Polycyclic presentation:
2^6,7|c^2=f,d^2=e,e^2=f,d^a=de,e^a=ef,g^a=g^6,d^b=df82 conjugacy classes — view the full character table.
| maximal subgroup of | #11726452, #11726457, #11726473, … |
| maximal normal subgroup of | #11726452, #11726457, #11726473, … |
| maximal quotient of | #11735924, #11735939, #11735942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1201);