Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4⋊C4×Dic7 |
| Sylow | C7 ⋊ C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C2×C7 | C2×C42 |
| Fitting | C2×C7×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C7 | C23 |
| hypercenter | C2×C4⋊C4 | D7 |
| hypocenter | C7 | C4×C4⋊C4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
194 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 194 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) = 19 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(4,5)(6,7)(12,13,14,15);(1,2,4,6,7,5,3)(16,17,18,19);(8,9,10,11)(17,19)100 conjugacy classes — view the full character table.
| maximal subgroup of | #11726534, #11726535, #11726536, … |
| maximal normal subgroup of | #11726534, #11726535, #11726536, … |
| maximal quotient of | Dic7×#95, #11727155, #11727161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 509);