Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C7 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C22×C7 | C22×C4 |
| Fitting | C7×#95 | C2 |
| socle, prime radical, cosocle | C23×C7 | C23 |
| hypercenter | #95 | D7 |
| hypocenter | C7 | #391 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,9,10,11)(13,15);(12,13)(14,15)(16,18,19,23)(17,20,21,22);(2,3)(4,5)(6,7)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^6,7|a^2=e,b^2=e,c^2=d,b^a=be,g^a=g^6,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89418, #11730779, #11730784, … |
| maximal normal subgroup of | C2×#89418, #11730779, #11730784, … |
| maximal quotient of | #11726373, #11726374, #11726484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 180);