Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator, socle, prime radical, cosocle | C23×C7 | C23 |
| Fitting | C2×C7×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D7 |
| hypocenter | C7 | #398 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
150 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 63 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)(9,11)(12,14,15,13);(13,14)(16,18,19,23)(17,20,21,22);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,14)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^6,7|a^2=e,b^2=ef,c^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 | #11731114, #11731115, #11731116, … |
| maximal normal subgroup of | #11731114, #11731115, #11731116, … |
| maximal quotient of | #11727164, #11727165, #11727168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 518);