Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#131 ⋊ C2 |
| Sylow | C7 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D28 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×#131 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | #131 | D7 |
| hypocenter | C7 | #578 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
248 conjugacy classes of subgroups.
| 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)(8,12,11,23,14,26,13,25)(9,17,27,37,18,32,16,24)(10,20,22,36,21,35,19,33)(15,29,28,38,31,39,30,34);(8,11,14,13)(9,16,18,27)(10,19,21,22)(12,23,26,25)(15,28,31,30)(17,24,32,37)(20,33,35,36)(29,38,39,34);(8,10,14,21)(9,15,18,31)(11,22,13,19)(12,20,26,35)(16,30,27,28)(17,29,32,39)(23,36,25,33)(24,34,37,38);(2,3)(4,5)(6,7)(8,9)(10,15)(11,16)(12,24)(13,27)(14,18)(17,25)(19,28)(20,34)(21,31)(22,30)(23,32)(26,37)(29,33)(35,38)(36,39)Polycyclic presentation:
2^6,7|b^2=f,c^2=f,d^2=ef,e^2=f,d^a=de,e^a=ef,g^a=g^6,c^b=cf82 conjugacy classes — view the full character table.
| maximal subgroup of | #11726881, #11726883, #11728849, … |
| maximal normal subgroup of | #11726881, #11726883, #11728849, … |
| maximal quotient of | #11735991, #11736007, #11736013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1206);