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Cayley diagram

C2×C5×D8 #4118

order160
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C4×#23882 (order 1024) · Inn(G) = D4 · Out(G) = C4×#120
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C23

Aliases

direct productC2×C5×D8

Subgroups

subgroupG/H
center, socleC22×C5D4
commutator, Frattini, cosocleC4C23×C5
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ G

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2)(11,12,13,14,15);(3,4,5,6,7,8,9,10);(4,10)(5,9)(6,8)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64782, #64783, C2×#3958, …
maximal subgroup of#64782, #64783, C5×#357, …
maximal normal subgroup of#64782, #64783, C5×#357, …
maximal quotient ofC2×C5×#102, C2×C5×#107, C4×C5×D8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 193);