Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#135 ⋊ C2 |
| Sylow | C7 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×#135 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | #135 | D7 |
| hypocenter | C7 | #578 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,4,6,7,5,3)(8,11,13,21)(9,16,14,28)(10,19,18,36)(12,25,26,23)(15,30,27,38)(17,35,29,33)(20,37,24,22)(31,39,34,32);(8,10,13,18)(9,15,14,27)(11,22,21,37)(12,24,26,20)(16,32,28,39)(17,34,29,31)(19,25,36,23)(30,35,38,33);(9,14)(10,18)(11,21)(17,29)(20,24)(23,25)(30,38)(32,39);(2,3)(4,5)(6,7)(8,9)(10,15)(11,16)(12,17)(13,14)(18,27)(19,30)(20,31)(21,28)(22,32)(23,33)(24,34)(25,35)(26,29)(36,38)(37,39)Polycyclic presentation:
2^6,7|c^2=f,d^2=f,e^2=f,b^a=bf,g^a=g^6,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #11726997, #11726999, #11729272, … |
| maximal normal subgroup of | #11726997, #11726999, #11729272, … |
| maximal quotient of | #11736266, #11736268, #11736273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1224);