Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #1141 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#6513 |
| Fitting | C2×C7×#41 ⋊ C2 |
| Sylow | C7 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×#41 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | C2×#135 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41 | |
266 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 25 points:
(3,4,6,8,9,7,5)(11,16)(13,21)(15,22)(17,18)(19,25)(23,24);(1,2)(10,12,14,20)(11,15,18,24)(13,19,21,25)(16,22,17,23);(4,5)(6,7)(8,9)(10,11)(12,15)(13,16)(14,18)(17,21)(19,22)(20,24)(23,25);(1,2)88 conjugacy classes — view the full character table.
| maximal subgroup of | #11727753, #11727754, #11727765, … |
| maximal normal subgroup of | #11727753, #11727754, #11727765, … |
| maximal quotient of | C2×#89770, C2×#89774, C2×#89775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1274);