Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×2−1+4 ⋊ C2 |
| Sylow | C5 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×2−1+4 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | 2−1+4 | D5 |
| hypocenter | C5 | #578 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
248 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,4,5,3)(7,14)(10,23)(12,25)(13,29)(15,16)(18,33)(21,35)(24,36)(26,27)(28,37)(30,31)(32,34);(6,9,11,22)(7,13,16,31)(8,17,19,20)(10,21,23,35)(12,24,27,28)(14,29,15,30)(18,32,33,34)(25,36,26,37);(6,8,11,19)(7,12,16,27)(9,20,22,17)(10,18,23,33)(13,28,31,24)(14,25,15,26)(21,34,35,32)(29,37,30,36);(2,3)(4,5)(6,7)(8,12)(9,13)(10,14)(11,16)(15,23)(17,24)(18,25)(19,27)(20,28)(21,29)(22,31)(26,33)(30,35)(32,36)(34,37)Polycyclic presentation:
2^6,5|b^2=f,c^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^4,c^b=cf,e^d=ef56 conjugacy classes — view the full character table.
| maximal subgroup of | #10612019, #10612020, #10612032, … |
| maximal normal subgroup of | #10612019, #10612020, #10612032, … |
| maximal quotient of | #10614278, #10614280, #10614281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1510);