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Cayley diagram

AGL(1,𝔽5)2 #87911

order400
isoclinismisoclinic class of AGL(1,𝔽5)2
automorphismAut(G) = AGL(1,𝔽5)2⋊C2 (order 800) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

direct productAGL(1,𝔽5)2

Splittings

Fitting, socle, derived, SylowC52 ⋊ C42

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52C42
prime radical, cosocleD52C22
LCSG ▷ C52
UCSC1

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2,3,5,4)(7,8,9,10);(2,3,4,5)(6,7,8,10,9)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)2⋊C2, C2×AGL(1,𝔽5)2, C3×AGL(1,𝔽5)2, …
maximal normal subgroup ofAGL(1,𝔽5)2⋊C2, C2×AGL(1,𝔽5)2, C3×AGL(1,𝔽5)2, …
maximal quotient of(C2×AGL(1,𝔽5))⋊AGL(1,𝔽5), AGL(1,𝔽5)×#170, #11717300, …
automorphism group of#1042, D5×AGL(1,𝔽5)
holomorph ofAGL(1,𝔽5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 205);