Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×#41 ⋊ C2 |
| Sylow | C7 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×#41 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | #575 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#41 | |
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)(9,15)(10,17)(12,21)(13,19)(14,22)(16,23)(18,20);(8,10,13,12)(9,14,16,20)(11,17,19,21)(15,22,23,18);(11,19)(15,23)(17,21)(18,22);(2,3)(4,5)(6,7)(8,9)(10,14)(11,18)(12,20)(13,16)(15,17)(19,22)(21,23)Polycyclic presentation:
2^6,7|c^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^6,d^b=df,e^d=ef82 conjugacy classes — view the full character table.
| maximal subgroup of | #11727766, #11727767, #11727797, … |
| maximal normal subgroup of | #11727766, #11727767, #11727797, … |
| maximal quotient of | #11739652, #11739658, #11739660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1275);