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Cayley diagram

C5×M16 #695

order80
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C4×D4 (order 64) · Inn(G) = C22 · Out(G) = C22×C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×#23872
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×M16

Subgroups

subgroupG/H
centerC4×C5C22
commutatorC2C2×C4×C5
Frattini, cosocleC4C22×C5
socleC2×C5C2×C4
prime radicalC22×C5C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ G

20 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#3954, #3955, #3956, …
maximal subgroup of#3954, #3955, #3956, …
maximal normal subgroup of#3954, #3955, #3956, …
maximal quotient ofC5×#97, C5×C22⋊C8, C5×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 24);