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Cayley diagram

C5×C22⋊C4 #692

order80
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C4×#120 (order 128) · Inn(G) = C22 · Out(G) = C4×D4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×#511790
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×C22⋊C4
unambiguous semidirect productC22⋊(C4×C5)

Subgroups

subgroupG/H
center, socleC22×C5C22
commutatorC2C2×C4×C5
Frattini, cosocleC22C22×C5
prime radicalC23×C5C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ G

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#3938, #4023, #4024, …
maximal subgroup of#3938, C5×#99, #4023, …
maximal normal subgroup of#3938, C5×#99, #4023, …
maximal quotient ofC5×#95, C5×C22⋊C8, C5×#99, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 21);