Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C4×C7×D4 | C2 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #553 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,11,14)(9,13,12,15)(16,19,20,18)(17,21,22,23);(9,12)(13,15)(16,18,20,19)(17,21,22,23);(10,14)(13,15);(2,3)(4,5)(6,7)(8,9,11,12)(10,13,14,15)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^6,7|a^2=f,c^2=e,d^2=f,c^a=cf,g^a=g^6,d^b=def85 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90231, #11742853, #11742855, … |
| maximal normal subgroup of | C2×#90231, #11742853, #11742855, … |
| maximal quotient of | #11730903, #11730906, #11730907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 993);