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Cayley diagram

C2×C4×C5 #176

order40
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C4×D4 (order 32) · Inn(G) = C1 · Out(G) = C4×D4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×B(4,𝔽2)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC2×C4×C5
multiplicative(ℤ/55)ˣ, (ℤ/75)ˣ, (ℤ/100)ˣ, (ℤ/110)ˣ, (ℤ/150)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C22×C5
socle, prime radicalC22×C5C2
LCSG ▷ C1
UCSC1 ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

center ofC5×#97, C5×C22⋊C8, C5×C4⋊C8, …
derived subgroup of#64671, #64672, #64684, …
Frattini subgroup ofC4×#6014, #11716351, C4⋊#6014, …
Fitting subgroup ofC2×#168, #681, C4×Dic5, …
maximal subgroup ofC2×#168, #681, C4×Dic5, …
maximal normal subgroup ofC2×#168, #681, C4×Dic5, …
maximal quotient ofC42×C5, C5×C22⋊C4, C5×C4⋊C4, …
automorphism group ofC5×C11, C3×C25, C4×C25, …
outer automorphism group ofC5×C11, C3×C25, C4×C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 9);