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Cayley diagram

C2×C11⋊C5 #1121

order110
isoclinismisoclinic class of C11⋊C5
automorphismAut(G) = AGL(1,𝔽11) (order 110) · Inn(G) = C11⋊C5 · Out(G) = C2
holomorphG ⋊ Aut(G) = C2×(C11⋊C5)⋊AGL(1,𝔽11)
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×C11⋊C5
unambiguous semidirect product(C2×C11)⋊C5

Splittings

Fitting, socle, HallC2×C11 ⋊ C5
derived, SylowC11 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC2C11⋊C5
commutator, cosocle, hypocenterC11C2×C5
FrattiniC1G
Fitting, socleC2×C11C5
LCSG ▷ C11
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 22

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2)(4,5,7,10,9)(6,8,12,13,11);(3,4,6,9,5,7,11,13,8,10,12)

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic11⋊C5, C4×C11⋊C5, C2×AGL(1,𝔽11), …
maximal normal subgroup ofDic11⋊C5, C4×C11⋊C5, C2×AGL(1,𝔽11), …
maximal quotient ofC4×C11⋊C5, C22×C11⋊C5, D3×C11⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(110, 2);