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Cayley diagram

C7×C11 #664

order77
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C5 (order 60) · Inn(G) = C1 · Out(G) = C22×C3×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽7)×AGL(1,𝔽11)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC77
abelianC7×C11

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 18 points:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

center ofD3×C7×C11, C7×D5×C11, C7×C11×A4, …
derived subgroup ofD77, Dic77, D7×D11, …
Frattini subgroup ofC3×C49×C121, C121×C49⋊C3, C22×C49×C121, …
Fitting subgroup ofC11×D7, C7×D11, D77, …
socle ofC11×D7, C7×D11, D77, …
maximal subgroup ofC11×D7, C7×D11, D77, …
maximal normal subgroup ofC11×D7, C7×D11, D77, …
maximal quotient ofC2×C7×C11, C3×C7×C11, C5×C7×C11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(77, 1);