Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#99 ⋊ C2 |
| Sylow | C5 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4028 |
| commutator, cosocle | C23×C5 | C23 |
| Frattini | C23 | C22×D5 |
| Fitting | C5×#99 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #99 | D5 |
| hypocenter | C5 | B(4,𝔽2) |
| LCS | G ▷ C23×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #99 | |
198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 198 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) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,5,3)(6,7)(8,12,13,10)(9,11);(6,8)(7,10)(9,13)(11,12);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)Polycyclic presentation:
2^6,5|c^2=d,c^a=cd,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 | #10599923, #10599924, #10599931, … |
| maximal normal subgroup of | #10599923, #10599924, #10599931, … |
| maximal quotient of | #10598899, #10598900, #10598901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 368);