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C7⋊(C11⋊AGL(1,𝔽5)) #13095366

order1540
isoclinismisoclinic class of C7⋊(C11⋊AGL(1,𝔽5))
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC7⋊(C11⋊AGL(1,𝔽5))
unambiguous semidirect productC11⋊(C7⋊AGL(1,𝔽5))

Splittings

Fitting, socle, derived, HallC5×C7×C11 ⋊ C4
HallC7×C11 ⋊ AGL(1,𝔽5)
HallC5×C11 ⋊ Dic7
HallC5×C7 ⋊ Dic11
SylowC11 ⋊ C7⋊AGL(1,𝔽5)
SylowC7 ⋊ C11⋊AGL(1,𝔽5)
SylowC5 ⋊ Dic77

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C11C4
prime radical, cosocleC7×D5×C11C2
LCSG ▷ C5×C7×C11
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 385

Polycyclic presentation:

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

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), C3×C7⋊(C11⋊AGL(1,𝔽5)), C3⋊(C7⋊(C11⋊AGL(1,𝔽5))), …
maximal normal subgroup ofC11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), C3×C7⋊(C11⋊AGL(1,𝔽5)), C11×C7⋊(C11⋊AGL(1,𝔽5)), …
maximal quotient ofC3×C7⋊(C11⋊AGL(1,𝔽5)), C3⋊(C7⋊(C11⋊AGL(1,𝔽5))), C7⋊(C11⋊#4380), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1540, 21);