Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #6400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#135 ⋊ C2 |
| Sylow | C7 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1141 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×#135 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | #135 | D7 |
| hypocenter | C7 | #508 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,4,6,7,5,3)(9,15)(11,22)(12,23)(14,26)(16,17)(18,31)(19,32)(20,33)(24,36)(25,30)(27,28)(29,38)(34,39)(35,37);(8,10,13,21)(9,14,18,29)(11,19,24,34)(12,20,23,33)(15,26,31,38)(16,27,30,37)(17,28,25,35)(22,32,36,39);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,25)(13,18)(16,22)(17,23)(19,26)(20,35)(21,29)(24,31)(27,32)(28,33)(30,36)(34,38)(37,39)Polycyclic presentation:
2^6,7|b^2=f,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^6,d^c=de,e^c=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #11726992, #11726993, #11728675, … |
| maximal normal subgroup of | #11726992, #11726993, #11728675, … |
| maximal quotient of | #11728402, #11728405, #11728407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 728);