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Cayley diagram

D5×A4 #1229

order120
isoclinismisoclinic class of D5×A4
automorphismAut(G) = AGL(1,𝔽5)×S4 (order 480) · Inn(G) = G · Out(G) = C22
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productD5×A4
unambiguous semidirect productC5⋊(C2×A4)
unambiguous semidirect product(C22×D5)⋊C3

Splittings

Fitting, socle, derivedC22×C5 ⋊ C2×C3
HallC22×D5 ⋊ C3
SylowC5 ⋊ C2×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C5C2×C3
LCSG ▷ C22×C5
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4⋊AGL(1,𝔽5), A4×AGL(1,𝔽5), D5×S4, …
maximal normal subgroup ofA4⋊AGL(1,𝔽5), A4×AGL(1,𝔽5), D5×S4, …
maximal quotient ofC5⋊(#41⋊C3), D5×SL(2,𝔽3), A4×Dic5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 39);