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C7⋊(C23⋊AGL(1,𝔽11)) #31799338

order17710
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC7⋊(C23⋊AGL(1,𝔽11))
unambiguous semidirect productC23⋊(C7⋊AGL(1,𝔽11))
unambiguous semidirect product(C11⋊C5)⋊D161
unambiguous semidirect productD1771⋊C5

Splittings

Fitting, socle, derived, HallC7×C11×C23 ⋊ C2×C5
HallC7×C23×C11⋊C5 ⋊ C2
HallD1771 ⋊ C5
HallC23×C11⋊C5 ⋊ D7
HallC7×C11⋊C5 ⋊ D23
HallC11×C23 ⋊ C5×D7
HallC7×C23 ⋊ AGL(1,𝔽11)
HallC7×C11 ⋊ C5×D23
HallC11⋊C5 ⋊ D161
SylowC23 ⋊ C7⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×D161
SylowC7 ⋊ C23⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C23C2×C5
LCSG ▷ C7×C11×C23
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1771

Polycyclic presentation:

2,5,7,11,23|c^a=c^6,d^a=d^10,e^a=e^22,d^b=d^4

Conjugacy classes & character table

571 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD23×C7⋊AGL(1,𝔽11), AGL(1,𝔽11)×D161, D7×C23⋊AGL(1,𝔽11), …
maximal normal subgroup ofD23×C7⋊AGL(1,𝔽11), AGL(1,𝔽11)×D161, D7×C23⋊AGL(1,𝔽11), …
maximal quotient ofC7⋊(C23⋊(Dic11⋊C5)), C2×C7⋊(C23⋊AGL(1,𝔽11))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17710, 32);