Finite Group DatabasebetaFinite Group Databasebeta| order | 7680 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C27 |
| direct product | D5×#11715850 |
| Fitting | C22×C5×H(5,𝔽2) ⋊ C2×D3 |
| derived | C5×C22⋊#997 ⋊ C22 |
| Sylow | C5 ⋊ C2×#11715850 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×#87588 |
| commutator, cosocle, hypocenter | C5×C22⋊#997 | C22 |
| Fitting | C22×C5×H(5,𝔽2) | C2×D3 |
| socle | C23×C5 | C2×#1020 |
| LCS | G ▷ C5×C22⋊#997 | |
| UCS | C1 ◁ C2 | |
18616 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 18616 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3,4)(6,11,9)(7,12,10)(14,15)(16,17);(1,2,4,3)(6,7)(10,11)(13,14,16,17,15);(5,6)(7,9)(8,10)(11,12)116 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#11715850 |
| maximal normal subgroup of | AGL(1,𝔽5)×#11715850 |
| maximal quotient of | D5×#72209641, D5×#72209772 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(6,11,9)(7,12,10)(14,15)(16,17), (1,2,4,3)(6,7)(10,11)(13,14,16,17,15), (5,6)(7,9)(8,10)(11,12));