Finite Group Databasebeta

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Cayley diagram

C4⋊#1035 #87737

order400
isoclinismisoclinic class of #87735
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊#1035

Splittings

FittingC4×C25 ⋊ C4
SylowC25 ⋊ C4⋊C4

Subgroups

subgroupG/H
centerC2C2×#1035
commutatorC2×C25C2×C4
Frattini, socleC2×C5C2×AGL(1,𝔽5)
FittingC4×C25C4
prime radical, cosocleC2×D25C22
hypercenterC4#1035
hypocenterC25C4⋊C4
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ C4

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

(1,7,12,10,5,3,19,17,15,13,8,25,24,16,22,20,21,6,11,14,9,2,4,23,18)(26,27,28,29);(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716411, #11716412, #11716424, …
maximal normal subgroup of#11716411, #11716412, #11716424, …
maximal quotient of#11716407, #11716408, #11716411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 31);