Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#125 ⋊ C2 |
| Sylow | C7 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#125 | C2 |
| prime radical | C22×D7 | C23 |
| hypercenter | #125 | D7 |
| hypocenter | C7 | #541 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #125 | |
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,4,6,7,5,3)(8,10,11,14)(9,13,12,15)(17,23)(21,29)(22,31)(28,30);(8,9)(10,13)(11,12)(14,15)(16,19,20,27)(17,22,23,31)(18,24,25,26)(21,28,29,30);(9,12)(10,14)(16,18,20,25)(17,21,23,29)(19,26,27,24)(22,30,31,28);(2,3)(4,5)(6,7)(9,12)(10,14)(16,17)(18,21)(19,22)(20,23)(24,28)(25,29)(26,30)(27,31)Polycyclic presentation:
2^6,7|b^2=e,c^2=e,d^2=f,c^a=cf,d^a=def,g^a=g^6,c^b=cef,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90387, #11743157, #11743167, … |
| maximal normal subgroup of | C2×#90387, #11743157, #11743167, … |
| maximal quotient of | #11731596, #11731600, #11731609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1149);