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Cayley diagram

C5⋊#401 #64951

order320
isoclinismisoclinic class of #64945
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect productC5⋊#401

Splittings

SylowC5 ⋊ #401

Subgroups

subgroupG/H
center, FrattiniC23C22×D5
commutator, socle, prime radical, cosocleC23×C5C23
FittingC5×#95C2
hypercenter#95D5
hypocenterC5#401
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #95

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

2^6,5|a^2=f,b^2=d,c^2=ef,b^a=be,c^a=cdef,g^a=g^4,c^b=cf

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C5⋊#401, #10605084, #10605087, …
maximal normal subgroup ofC2×C5⋊#401, #10605084, #10605087, …
maximal quotient ofC5⋊#2104, #10600209, #10600210, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 289);