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C23×C4⋊AGL(1,𝔽11) #72248869

order3520
automorphism?
homologyAbelianization (H1) = C25×C5 · Schur multiplier (H2) = C210

Aliases

direct productC23×C4⋊AGL(1,𝔽11)

Splittings

FittingC23×C4×C11 ⋊ C2×C5
HallC23×D44 ⋊ C5
HallC11⋊C5 ⋊ C23×D4
SylowC11 ⋊ C23×C5×D4

Subgroups

subgroupG/H
centerC24C2×AGL(1,𝔽11)
commutator, cosocleC2×C11C25×C5
FrattiniC2C24×AGL(1,𝔽11)
FittingC23×C4×C11C2×C5
socleC24×C11C22×C5
hypercenterC23×C4AGL(1,𝔽11)
hypocenterC11C23×C5×D4
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C24 ◁ C23×C4

2724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 352

A minimal faithful permutation representation — 5 generators on 21 points:

(5,6)(11,12,13,18,21)(14,19,15,20,17);(3,4);(7,8,9,10);(1,2);(8,10)(12,13)(14,15)(16,17)(18,20)(19,21)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72248961, C23×D4×AGL(1,𝔽11), C22×#66935⋊C5
maximal normal subgroup of#72248961, C23×D4×AGL(1,𝔽11), C22×#66935⋊C5
maximal quotient of#72254484, C22×#66935⋊C5, #72269486, …

GAP

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

gap> G := Group((5,6)(11,12,13,18,21)(14,19,15,20,17), (3,4), (7,8,9,10), (1,2), (8,10)(12,13)(14,15)(16,17)(18,20)(19,21));