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ΓL(1,𝔽16)⋊Dic3 #10623977

order720
isoclinismisoclinic class of ΓL(1,𝔽16)⋊D3
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect productΓL(1,𝔽16)⋊Dic3

Splittings

derived, HallC32×C5 ⋊ C42
SylowC32 ⋊ C4×AGL(1,𝔽5)
SylowC5 ⋊ Dic32

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽16)⋊D3
commutator, hypocenterC32×C5C42
Fitting, socleC2×C32×C5C2×C4
prime radical, cosocleC2×C32×D5C22
LCSG ▷ C32×C5
UCSC1 ◁ C2

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 120

Polycyclic presentation:

2^4,3^2,5|a^2=c,b^2=d,e^a=e^2,g^a=g^3,e^b=e^2,f^b=f^2,g^c=g^4

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13088358, #13088359, D3×Dic3×AGL(1,𝔽5), …
maximal normal subgroup of#13088358, #13088359, D3×Dic3×AGL(1,𝔽5), …
maximal quotient ofΓL(1,𝔽16)⋊(C3⋊C8), #13085474, #13085487, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 446);