Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C5 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C22×C5 | C22×C4 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | #387 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
154 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(10,11,12,13)(15,19)(18,21);(11,13)(14,16,17,20)(15,18,19,21);(2,3)(4,5)(6,7,8,9)(14,15)(16,18)(17,19)(20,21)Polycyclic presentation:
2^6,5|a^2=d,b^2=e,c^2=f,c^a=ce,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4044, C4×#4035, #10605360, … |
| maximal normal subgroup of | C4×#4044, C4×#4035, #10605360, … |
| maximal quotient of | #10600194, #10600200, #10600205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 608);