Finite Group Databasebeta

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Cayley diagram

Dic5×AGL(1,𝔽5) #87823

order400
isoclinismisoclinic class of D5×AGL(1,𝔽5)
automorphismAut(G) = C22×AGL(1,𝔽5)2 (order 1600) · Inn(G) = D5×AGL(1,𝔽5) · Out(G) = C23
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

direct productDic5×AGL(1,𝔽5)

Splittings

derived, SylowC52 ⋊ C42

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×AGL(1,𝔽5)
commutator, hypocenterC52C42
Fitting, socleC2×C52C2×C4
prime radical, cosocleC2×C5×D5C22
LCSG ▷ C52
UCSC1 ◁ C2

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,4,5,3)(7,8,9,10);(2,3)(4,5)(6,7,8,10,9)(11,12,13,14)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#170, #11717301, AGL(1,𝔽5)×Dic10, …
maximal normal subgroup ofAGL(1,𝔽5)×#170, #11717301, AGL(1,𝔽5)×Dic10, …
maximal quotient ofAGL(1,𝔽5)×#168, #11716731, #11716732, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 117);