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C7⋊((C11⋊C5)⋊AGL(1,𝔽5)) #21440397

order7700
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC7⋊((C11⋊C5)⋊AGL(1,𝔽5))
unambiguous semidirect product(C11⋊C5)⋊(C7⋊AGL(1,𝔽5))
unambiguous semidirect product(C7⋊(C11⋊AGL(1,𝔽5)))⋊C5

Splittings

Fitting, socle, derivedC5×C7×C11 ⋊ C4×C5
HallC5×C7×C11⋊C5 ⋊ C4
HallC5×C11⋊C5 ⋊ Dic7
HallC7×C11 ⋊ C5×AGL(1,𝔽5)
SylowC11 ⋊ C5×C7⋊AGL(1,𝔽5)
SylowC7 ⋊ (C11⋊C5)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C11C4×C5
LCSG ▷ C5×C7×C11
UCSC1

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊((C11⋊C5)⋊AGL(1,𝔽5)), C3×C7⋊((C11⋊C5)⋊AGL(1,𝔽5)), C3⋊(C7⋊((C11⋊C5)⋊AGL(1,𝔽5)))
maximal normal subgroup ofΓL(1,𝔽8)⋊((C11⋊C5)⋊AGL(1,𝔽5)), C3×C7⋊((C11⋊C5)⋊AGL(1,𝔽5))
maximal quotient ofC3×C7⋊((C11⋊C5)⋊AGL(1,𝔽5)), C3⋊(C7⋊((C11⋊C5)⋊AGL(1,𝔽5)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7700, 61);