Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C5×C16 ⋊ C2 |
| Sylow | C5 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D5 |
| commutator, socle | C2×C5 | C22×C8 |
| Frattini | C8 | C22×D5 |
| Fitting | C2×C5×C16 | C2 |
| prime radical | #709 | C4 |
| cosocle | C5×C8 | C23 |
| hypercenter | C2×C16 | D5 |
| hypocenter | C5 | #504 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C16 ◁ C2×C16 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,4,5,3)(6,8,9,16,10,17,19,28,11,18,20,29,21,30,31,36)(7,13,14,25,15,26,27,35,12,22,23,32,24,33,34,37);(7,12)(13,22)(14,23)(15,24)(25,32)(26,33)(27,34)(35,37);(2,3)(4,5)(6,7)(8,13)(9,14)(10,15)(11,12)(16,25)(17,26)(18,22)(19,27)(20,23)(21,24)(28,35)(29,32)(30,33)(31,34)(36,37)Polycyclic presentation:
2^6,5|c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^4104 conjugacy classes — view the full character table.
| maximal subgroup of | #10598086, #10598094, #10599849, … |
| maximal normal subgroup of | #10598086, #10598094, #10599849, … |
| maximal quotient of | C16×Dic10, #10599841, #10599844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 528);