Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #394 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator, socle, cosocle | C23×C7 | C23 |
| Fitting, prime radical | C22×C7×D4 | C2 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | #394 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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)(9,11)(16,17)(18,19);(13,15)(17,19);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17,18,19)Polycyclic presentation:
2^6,7|a^2=f,b^a=be,c^a=cd,g^a=g^6,c^b=cf82 conjugacy classes — view the full character table.
| maximal subgroup of | #11728724, #11728725, #11728732, … |
| maximal normal subgroup of | #11728724, #11728725, #11728732, … |
| maximal quotient of | #11729301, #11729303, #11729320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 756);