Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #385 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C22×C7 | C22×C4 |
| Fitting | C23×C4×C7 | C2 |
| socle, cosocle | C23×C7 | C23 |
| prime radical | C24×C7 | C22 |
| hypercenter | C23×C4 | D7 |
| hypocenter | C7 | #385 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
234 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(8,9,10,11)(17,19);(13,15);(2,3)(4,5)(6,7)(12,13)(14,15)(16,17,18,19)Polycyclic presentation:
2^6,7|a^2=f,c^2=d,b^a=be,c^a=cf,g^a=g^6124 conjugacy classes — view the full character table.
| maximal subgroup of | #11727348, #11727349, #11727351, … |
| maximal normal subgroup of | #11727348, #11727349, #11727351, … |
| maximal quotient of | #11729290, #11729301, #11729309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 744);