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Cayley diagram

C2×C32×AGL(1,𝔽5) #67215

order360
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C32×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C32×C4)

Splittings

Fitting, socleC2×C32×C5 ⋊ C4
derived, SylowC5 ⋊ C2×C32×C4
HallC32×C5 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C32AGL(1,𝔽5)
commutator, hypocenterC5C2×C32×C4
FrattiniC1G
Fitting, socleC2×C32×C5C4
prime radicalC2×C32×D5C2
cosocleD5C22×C32
LCSG ▷ C5
UCSC1 ◁ C2×C32

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 2 generators on 13 points:

(3,4,5)(10,11,12,13);(1,2)(6,7,8)(9,10,11,13,12)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×(C2×D3)⋊AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), …
maximal normal subgroup ofC3×Dic3×AGL(1,𝔽5), C3×(C2×D3)⋊AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), …
maximal quotient ofC32×D5⋊C8, C32×#700, C32×C4×AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 145);