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Cayley diagram

C5×C4⋊C4 #693

order80
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C4×#120 (order 128) · Inn(G) = C22 · Out(G) = C4×D4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×C4⋊C4
unambiguous semidirect productC4⋊(C4×C5)

Subgroups

subgroupG/H
center, socle, prime radicalC22×C5C22
commutatorC2C2×C4×C5
Frattini, cosocleC22C22×C5
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ G

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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#3939, #3940, #3941, …
maximal subgroup of#3939, #3940, #3941, …
maximal normal subgroup of#3939, #3940, #3941, …
maximal quotient ofC5×#95, C5×C4⋊C8, C5×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 22);