Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#104 ⋊ C2 |
| Sylow | C5 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4028 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#104 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #104 | D5 |
| hypocenter | C5 | #454 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #104 | |
146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 3 generators on 21 points:
(1,2,4,5,3)(6,9,20,8,11,17,10,18)(7,13,15,21,16,12,14,19);(6,8,11,18)(7,12,16,13)(9,20,17,10)(14,21,15,19);(2,3)(4,5)(6,7)(8,12)(9,19)(10,15)(11,16)(13,18)(14,20)(17,21)Polycyclic presentation:
2^6,5|b^2=f,c^2=df,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^4,c^b=ce,e^b=ef,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10610717, C2×#65116, #10610725, … |
| maximal normal subgroup of | #10610717, C2×#65116, #10610725, … |
| maximal quotient of | #10598973, #10598974, #10598980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 454);