Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle, prime radical | C22×C7 | C22×C4 |
| Fitting | C7×C4⋊C8 | C2 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C4⋊C8 | D7 |
| hypocenter | C7 | #446 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(10,14)(12,15)(16,18,19,21,20,22,23,17);(8,10,11,14)(9,12,13,15);(2,3)(4,5)(6,7)(8,9,11,13)(10,12,14,15)(16,17,19,18,20,21,23,22)Polycyclic presentation:
2^6,7|a^2=e,b^2=d,c^2=de,e^2=f,c^a=cf,g^a=g^6,c^b=cd88 conjugacy classes — view the full character table.
| maximal subgroup of | #11725908, #11725910, #11725915, … |
| maximal normal subgroup of | #11725908, #11725910, #11725915, … |
| maximal quotient of | #11726366, #11726374, #11726407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 365);