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C2×(C11⋊C5)⋊AGL(1,𝔽5) #72228526

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

Aliases

direct productC2×(C11⋊C5)⋊AGL(1,𝔽5)

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, hypercenterC2(C11⋊C5)⋊AGL(1,𝔽5)
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:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228527, C2×AGL(1,𝔽5)×AGL(1,𝔽11), #72228557, …
maximal normal subgroup ofC2×AGL(1,𝔽5)×AGL(1,𝔽11), C22×(C11⋊C5)⋊AGL(1,𝔽5), AGL(1,𝔽5)×Dic11⋊C5, …
maximal quotient of#72228527, #72228557, D3×(C11⋊C5)⋊AGL(1,𝔽5), …

GAP

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

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