Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×C4⋊C8 ⋊ C2 |
| Sylow | C7 ⋊ #424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C7×C4⋊C8 | C2 |
| prime radical | C22×D7 | C2×C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C4⋊C8 | D7 |
| hypocenter | C7 | #424 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
114 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,7,5,3)(9,11)(12,14,15,20,16,21,22,26)(13,18,19,25,17,23,24,27);(8,9,10,11)(13,17)(18,23)(19,24)(25,27);(2,3)(4,5)(6,7)(12,13)(14,18)(15,19)(16,17)(20,25)(21,23)(22,24)(26,27)Polycyclic presentation:
2^6,7|b^2=e,c^2=d,d^2=f,b^a=bf,g^a=g^6,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89611, #11736457, #11736461, … |
| maximal normal subgroup of | C2×#89611, #11736457, #11736461, … |
| maximal quotient of | #11726286, #11726287, #11726288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 373);