Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#126 ⋊ C2 |
| Sylow | C5 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#126 | C2 |
| prime radical | C2×C22⋊D5 | C22 |
| hypercenter | #126 | D5 |
| hypocenter | C5 | #563 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #126 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4,5,3)(6,8,9,12)(7,11,10,13)(15,21)(16,23)(17,25)(26,28);(7,10)(11,13)(14,17)(15,20)(16,22)(18,25)(19,26)(21,24)(23,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,16,18,23)(15,19,21,27)(17,22,25,29)(20,26,24,28);(2,3)(4,5)(8,12)(11,13)(14,15)(16,19)(17,24)(18,21)(20,25)(22,28)(23,27)(26,29)Polycyclic presentation:
2^6,5|b^2=f,d^2=e,c^a=cf,d^a=def,g^a=g^4,c^b=ce,d^b=df,d^c=df47 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66046, #10618011, #10618019, … |
| maximal normal subgroup of | C2×#66046, #10618011, #10618019, … |
| maximal quotient of | #10605996, #10606011, #10606012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1384);