Finite Group Databasebeta

← Prev (#6841) Next (#6843) →

Cayley diagram

C22×C5×A4 #6842

order240
isoclinismisoclinic class of A4
automorphismAut(G) = C4×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22

Aliases

direct productC22×C5×A4

Splittings

Fitting, socle, HallC24×C5 ⋊ C3
derivedC22 ⋊ C22×C3×C5
SylowC24 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC22×C5A4
commutator, cosocle, hypocenterC22C22×C3×C5
FrattiniC1G
Fitting, socleC24×C5C3
LCSG ▷ C22
UCSC1 ◁ C22×C5

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×C5⋊S4, #14511782, #14511783, …
maximal subgroup ofC2×C5×SL(2,ℤ/4), C5×GL(2,ℤ/4), C2×A4⋊Dic5, …
maximal normal subgroup ofC2×C5×SL(2,ℤ/4), C5×GL(2,ℤ/4), C2×A4⋊Dic5, …
maximal quotient ofC2×C4×C5×A4, C5×D4×A4, C22×C5×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 203);