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C2×AGL(1,𝔽5)×C11⋊C5 #72228513

order2200
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽5)×C11⋊C5

Splittings

Fitting, socleC2×C5×C11 ⋊ C4×C5
derivedC5×C11 ⋊ C2×C4×C5
HallC5×C11⋊C5 ⋊ C2×C4
SylowC11 ⋊ C2×C5×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽5)×C11⋊C5
commutator, hypocenterC5×C11C2×C4×C5
FrattiniC1G
Fitting, socleC2×C5×C11C4×C5
prime radicalC2×D5×C11⋊C5C2
cosocleD5×C11C22×C5
LCSG ▷ C5×C11
UCSC1 ◁ C2

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 110

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11⋊C5×D5⋊AGL(1,𝔽5), C2×AGL(1,𝔽5)×AGL(1,𝔽11), AGL(1,𝔽5)×#11925895, …
maximal normal subgroup ofC2×AGL(1,𝔽5)×AGL(1,𝔽11), C22×AGL(1,𝔽5)×C11⋊C5, C4×AGL(1,𝔽5)×C11⋊C5, …
maximal quotient ofC11⋊C5×D5⋊AGL(1,𝔽5), AGL(1,𝔽5)×#11925895, D3×AGL(1,𝔽5)×C11⋊C5, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,5,8,6)(4,7,10,9,11)(12,13,14,15), (1,2,4,3,6,8,7,10,11,5,9)(13,14)(15,16)(17,18));