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AGL(1,𝔽5)2⋊C2 #11717532

order800
isoclinismisoclinic class of AGL(1,𝔽5)2⋊C2
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productAGL(1,𝔽5)2⋊C2
unambiguous semidirect product(#6053⋊C2)⋊C2

Splittings

Fitting, socle, SylowC52 ⋊ #104

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator#1044C2×C4
Fitting, socle, hypocenterC52#104
prime radical(D5⋊AGL(1,𝔽5))⋊C2C2
cosocleD5⋊AGL(1,𝔽5)C22
LCSG ▷ #1044 ▷ C5⋊D5 ▷ C52
UCSC1

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 10

Polycyclic presentation:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽5)2⋊C2, #72210663
maximal normal subgroup ofC2×AGL(1,𝔽5)2⋊C2
maximal quotient of#13107437, #13107438, #13107439, …
automorphism group of#1044, D52, D5⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(800, 1191);