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Cayley diagram

D5×#168 #87766

order400
isoclinismisoclinic class of D52
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productD5×#168

Splittings

derived, SylowC52 ⋊ C2×C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC4D52
commutator, hypocenterC52C2×C8
Fitting, cosocleC4×C52C22
socleC2×C52C2×C4
prime radicalC2×C5×D5C4
LCSG ▷ C52
UCSC1 ◁ C4

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,5,4,6,7,8)(9,10,12,13,11)(15,16)(17,18);(10,11)(12,13)(14,15,17,18,16)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#168, #11716731, #11716732, …
maximal normal subgroup ofAGL(1,𝔽5)×#168, #11716731, #11716732, …
maximal quotient ofD5×#672, #11716584, Dic5×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 60);