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Cayley diagram

C2×C3×AGL(1,𝔽5) #1230

order120
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C22×AGL(1,𝔽5) (order 80) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22
holomorphG ⋊ Aut(G) = D3×#13107468
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C3×C4)

Splittings

Fitting, socleC2×C3×C5 ⋊ C4
derived, SylowC5 ⋊ C2×C3×C4
HallC3×C5 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,𝔽5)
commutator, hypocenterC5C2×C3×C4
FrattiniC1G
Fitting, socleC2×C3×C5C4
prime radicalC2×C3×D5C2
cosocleD5C22×C3
LCSG ▷ C5
UCSC1 ◁ C2×C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 30

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×AGL(1,𝔽5), (C2×D3)⋊AGL(1,𝔽5), Dic3⋊AGL(1,𝔽5), …
maximal normal subgroup ofDic3×AGL(1,𝔽5), (C2×D3)⋊AGL(1,𝔽5), Dic3⋊AGL(1,𝔽5), …
maximal quotient ofC3×D5⋊C8, C3×#700, C3×C4×AGL(1,𝔽5), …
automorphism group ofC7×D5, C9×D5, C7×AGL(1,𝔽5), …
outer automorphism group ofC2×C7×5−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 40);