Finite Group Databasebeta

← Prev (#87593) Next (#87595) →

Cayley diagram

C7×C11⋊C5 #87594

order385
isoclinismisoclinic class of C11⋊C5
automorphismAut(G) = C2×C3×AGL(1,𝔽11) (order 660) · Inn(G) = C11⋊C5 · Out(G) = C22×C3
homologyAbelianization (H1) = C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C11⋊C5
unambiguous semidirect productC11⋊(C5×C7)
unambiguous semidirect product(C7×C11)⋊C5

Splittings

Fitting, socle, HallC7×C11 ⋊ C5
derived, SylowC11 ⋊ C5×C7

Subgroups

subgroupG/H
center, hypercenterC7C11⋊C5
commutator, cosocle, hypocenterC11C5×C7
FrattiniC1G
Fitting, socleC7×C11C5
LCSG ▷ C11
UCSC1 ◁ C7

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 77

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5,6,7)(9,10,12,15,14)(11,13,17,18,16);(8,9,11,14,10,12,16,18,13,15,17)

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×AGL(1,𝔽11), D7×C11⋊C5, C7⋊AGL(1,𝔽11), …
maximal normal subgroup ofC7×AGL(1,𝔽11), D7×C11⋊C5, C7⋊AGL(1,𝔽11), …
maximal quotient ofC2×C7×C11⋊C5, C3×C7×C11⋊C5, C7×C11⋊C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(385, 1);