Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×#104 ⋊ C2 |
| Sylow | C7 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6445 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#104 | C2 |
| socle | C2×C7 | #120 |
| hypercenter | #104 | D7 |
| hypocenter | C7 | #454 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #104 | |
146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,22,10,13,19,12,20)(9,15,17,23,18,14,16,21);(8,10,13,20)(9,14,18,15)(11,22,19,12)(16,23,17,21);(2,3)(4,5)(6,7)(8,9)(10,14)(11,21)(12,17)(13,18)(15,20)(16,22)(19,23)Polycyclic presentation:
2^6,7|b^2=f,c^2=df,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^6,c^b=ce,e^b=ef,e^c=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | #11736431, C2×#89599, #11736439, … |
| maximal normal subgroup of | #11736431, C2×#89599, #11736439, … |
| maximal quotient of | #11725934, #11725935, #11725941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 361);