Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×2−1+4 ⋊ C2 |
| Sylow | C7 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×2−1+4 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | 2−1+4 | D7 |
| hypocenter | C7 | #578 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
248 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,4,6,7,5,3)(9,16)(12,25)(14,27)(15,31)(17,18)(20,35)(23,37)(26,38)(28,29)(30,39)(32,33)(34,36);(8,11,13,24)(9,15,18,33)(10,19,21,22)(12,23,25,37)(14,26,29,30)(16,31,17,32)(20,34,35,36)(27,38,28,39);(8,10,13,21)(9,14,18,29)(11,22,24,19)(12,20,25,35)(15,30,33,26)(16,27,17,28)(23,36,37,34)(31,39,32,38);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,16)(13,18)(17,25)(19,26)(20,27)(21,29)(22,30)(23,31)(24,33)(28,35)(32,37)(34,38)(36,39)Polycyclic presentation:
2^6,7|b^2=f,c^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^6,c^b=cf,e^d=ef73 conjugacy classes — view the full character table.
| maximal subgroup of | #11737733, #11737734, #11737746, … |
| maximal normal subgroup of | #11737733, #11737734, #11737746, … |
| maximal quotient of | #11739992, #11739994, #11739995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1291);