Finite Group Databasebeta

← Prev (#92546) Next (#92548) →

Cayley diagram

C2×#6953 #92547

order500
isoclinismisoclinic class of #6953
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×#6953

Splittings

FittingC2×H(3,𝔽5) ⋊ C2
derived, SylowH(3,𝔽5) ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC2×C5C5⋊D5
commutator, cosocle, hypocenterH(3,𝔽5)C22
FrattiniC5C2×C5⋊D5
FittingC2×H(3,𝔽5)C2
LCSG ▷ H(3,𝔽5)
UCSC1 ◁ C2×C5

75 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 50

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11758660, #11758661, C4×#6953, …
maximal normal subgroup of#11758660, #11758661, C4×#6953, …
maximal quotient of#11758685, C4×#6953, C4⋊#6953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 33);