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Cayley diagram

C5×C4⋊C8 #3980

order160
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C4×#120 (order 256) · Inn(G) = C22 · Out(G) = C2×C4×D4
homologyAbelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2

Aliases

direct productC5×C4⋊C8
unambiguous semidirect productC4⋊(C5×C8)

Subgroups

subgroupG/H
centerC2×C4×C5C22
commutatorC2C2×C5×C8
Frattini, cosocleC2×C4C22×C5
socle, prime radicalC22×C5C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 17 points:

(2,4)(5,6,7,9,8,10,11,12)(13,14,15,16,17);(1,2,3,4)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#64699, #64700, #64701, …
maximal subgroup of#64699, #64700, #64701, …
maximal normal subgroup of#64699, #64700, #64701, …
maximal quotient ofC5×#334, C5×#335, C5×#336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 55);