Finite Group Databasebeta

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Cayley diagram

D5⋊#168 #87848

order400
isoclinismisoclinic class of #1042
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD5⋊#168

Splittings

derived, SylowC52 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC4#1042
commutator, hypocenterC52C2×C8
FrattiniC2C2×#1042
FittingC4×C52C4
socleC2×C52C2×C4
prime radicalC2×C5×D5C4
cosocleC5×Dic5C22
LCSG ▷ C52
UCSC1 ◁ C4

47 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:

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

Polycyclic presentation:

2^4,5^2|a^2=c,b^2=d,c^2=d,e^a=e^4,f^a=f^3,f^c=f^4

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#168, #11716732, #11716734, …
maximal normal subgroup ofAGL(1,𝔽5)×#168, #11716732, #11716734, …
maximal quotient ofD5⋊#672, #11716833, C4×#6032, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 142);