Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C5×#97 ⋊ C2 |
| Sylow | C5 ⋊ #433 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×#97 | C2 |
| prime radical | C22×D5 | C2×C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | #97 | D5 |
| hypocenter | C5 | #433 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #97 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4,5,3)(7,11)(10,13)(14,16,17,23,18,22,24,28)(15,20,21,27,19,25,26,29);(6,8,9,12)(7,10,11,13)(15,19)(16,22)(21,26)(23,28);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,20)(17,21)(18,19)(22,25)(23,27)(24,26)(28,29)Polycyclic presentation:
2^6,5|b^2=e,c^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | #10610212, C2×#64998, #10610220, … |
| maximal normal subgroup of | #10610212, C2×#64998, #10610220, … |
| maximal quotient of | #10599399, #10599401, #10599448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 336);