Finite Group Databasebeta

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Cayley diagram

C4⋊#1039 #87807

order400
isoclinismisoclinic class of #6045
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C5

Aliases

unambiguous semidirect productC4⋊#1039

Splittings

SylowC52 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×C5⋊D5
commutatorC2×C52C2×C4
FittingC2×C4×C52C2
socle, prime radical, cosocleC22×C52C22
hypercenterC2×C4C5⋊D5
hypocenterC52C4⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×C4

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 400

A minimal faithful permutation representation — 3 generators on 18 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716604, #11716607, #11716614, …
maximal normal subgroup of#11716604, #11716607, #11716614, …
maximal quotient ofC4⋊#6029, #11716684, #11716685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 101);