Finite Group Databasebeta

← Prev (#6838) Next (#6840) →

Cayley diagram

C22×C3×AGL(1,𝔽5) #6839

order240
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×S4
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C2×C3×C4)

Splittings

Fitting, socleC22×C3×C5 ⋊ C4
derived, SylowC5 ⋊ C22×C3×C4
HallC3×C5 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC22×C3AGL(1,𝔽5)
commutator, hypocenterC5C22×C3×C4
FrattiniC1G
Fitting, socleC22×C3×C5C4
prime radicalC22×C3×D5C2
cosocleD5C23×C3
LCSG ▷ C5
UCSC1 ◁ C22×C3

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊Dic3, C3×(C2×AGL(1,𝔽5))⋊C4, C2×Dic3×AGL(1,𝔽5), …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊Dic3, C3×(C2×AGL(1,𝔽5))⋊C4, C2×Dic3×AGL(1,𝔽5), …
maximal quotient ofC2×C3×D5⋊C8, C2×C3×#700, C3×D5⋊M16, …
automorphism group ofC7×Dic5, C2×C7×D5, C9×Dic5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 200);