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Cayley diagram

C4⋊Dic5 #684

order80
isoclinismisoclinic class of Dic10
automorphismAut(G) = AGL(1,𝔽5)×#120 (order 640) · Inn(G) = C2×D5 · Out(G) = C4×D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊Dic5

Splittings

SylowC5 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×D5
commutatorC2×C5C2×C4
FittingC2×C4×C5C2
socle, prime radical, cosocleC22×C5C22
hypercenterC2×C4D5
hypocenterC5C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 80

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

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

Polycyclic presentation:

2^4,5|a^2=c,b^2=d,b^a=bd,e^a=e^4

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#3939, #3940, #3948, …
maximal normal subgroup of#3939, #3940, #3948, …
maximal quotient ofC4⋊#168, #3949, #3950, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 13);