Finite Group Databasebeta

← Prev (#4115) Next (#4117) →

Cayley diagram

C2×C5×M16 #4116

order160
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C4×B(4,𝔽2) (order 512) · Inn(G) = C22 · Out(G) = C22×C4×D4
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC2×C5×M16

Subgroups

subgroupG/H
centerC2×C4×C5C22
commutatorC2C22×C4×C5
Frattini, cosocleC4C23×C5
socleC22×C5C2×C4
prime radicalC23×C5C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

68 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);(4,8)(6,10)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11986788, #11986789, #11986790, …
Fitting subgroup of#64773, #64774, #64775, …
maximal subgroup of#64773, #64774, #64775, …
maximal normal subgroup of#64773, #64774, #64775, …
maximal quotient ofC2×C5×#97, C4×C5×M16, C2×C5×C22⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 191);