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GO+(4,𝔽2)⋊C8 #10593153

order576
isoclinismisoclinic class of AΓL(1,𝔽9)
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect productGO+(4,𝔽2)⋊C8

Splittings

SylowC32 ⋊ D4⋊C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC4AΓL(1,𝔽9)
commutator#157C2×C8
FittingC32×C4SD16
socleC2×C32#102
prime radicalC2×GO+(4,𝔽2)C4
cosocleC4×#157C22
hypocenterC32D4⋊C8
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C4

109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 48

Polycyclic presentation:

2^6,3^2|a^2=c,c^2=ef,d^2=f,b^a=bcd,g^a=g^2h^2,h^a=g^2,c^b=ce,g^b=gh^2,h^b=h^2,g^c=g^2h,h^c=gh,g^d=g^2,h^d=h^2,g^e=g^2,h^e=h^2

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×GO+(4,𝔽2)⋊C8, #11774382, GO+(4,𝔽2)⋊M16, …
maximal normal subgroup ofC2×GO+(4,𝔽2)⋊C8, #11774382, GO+(4,𝔽2)⋊M16, …
maximal quotient of#11770663, #11770682, #11770802, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 5035);