Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#101 ⋊ C2 |
| Sylow | C5 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4028 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#101 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #101 | D5 |
| hypocenter | C5 | #454 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #101 | |
146 conjugacy classes of subgroups.
| 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)(6,9,17,7,11,18,10,13)(8,14,19,21,16,20,15,12);(6,8,11,16)(7,12,13,21)(9,14,18,20)(10,19,17,15);(2,3)(4,5)(6,7)(8,12)(9,17)(10,18)(11,13)(14,15)(16,21)(19,20)Polycyclic presentation:
2^6,5|b^2=f,c^2=df,d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^4,c^b=ce,d^b=df,e^c=ef38 conjugacy classes — view the full character table.
| maximal subgroup of | #10598575, #10599953, #10599954, … |
| maximal normal subgroup of | #10598575, #10599953, #10599954, … |
| maximal quotient of | #10598923, #10598924, #10598929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 380);