Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D20 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#131 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #131 | D5 |
| hypocenter | C5 | #577 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 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) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,9,14,10,15,16,20)(7,12,13,19,11,17,18,21);(7,11)(12,17)(13,18)(19,21);(6,7)(8,12)(9,13)(10,11)(14,19)(15,17)(16,18)(20,21);(2,3)(4,5)(8,14)(9,16)(12,19)(13,18)(15,20)(17,21)Polycyclic presentation:
2^6,5|d^2=e,e^2=f,d^a=de,e^a=ef,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10611623, #10611624, #10611631, … |
| maximal normal subgroup of | #10611623, #10611624, #10611631, … |
| maximal quotient of | #10610276, #10610291, #10610294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1423);