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C3⋊(C23⋊AGL(1,𝔽5)) #13063410

order1380
isoclinismisoclinic class of C3⋊(C23⋊AGL(1,𝔽5))
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C23⋊AGL(1,𝔽5))
unambiguous semidirect productC23⋊ΓL(1,𝔽16)

Splittings

Fitting, socle, derived, HallC3×C5×C23 ⋊ C4
HallC5×C23 ⋊ Dic3
HallC3×C23 ⋊ AGL(1,𝔽5)
SylowC23 ⋊ ΓL(1,𝔽16)
HallC3×C5 ⋊ Dic23
SylowC5 ⋊ Dic69
SylowC3 ⋊ C23⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C23C4
prime radical, cosocleC3×D5×C23C2
LCSG ▷ C3×C5×C23
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 345

Polycyclic presentation:

2^2,3,5,23|a^2=b,c^a=c^2,d^a=d^2,e^a=e^22,d^b=d^4

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC9⋊(C23⋊AGL(1,𝔽5)), C3×C3⋊(C23⋊AGL(1,𝔽5)), C3⋊(C3⋊(C23⋊AGL(1,𝔽5))), …
maximal normal subgroup ofC3×C3⋊(C23⋊AGL(1,𝔽5)), C3⋊((C23⋊C11)⋊AGL(1,𝔽5)), C11×C3⋊(C23⋊AGL(1,𝔽5)), …
maximal quotient ofC9⋊(C23⋊AGL(1,𝔽5)), C3×C3⋊(C23⋊AGL(1,𝔽5)), C3⋊(C3⋊(C23⋊AGL(1,𝔽5))), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 13);