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Cayley diagram

C11×C16 #4311

order176
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C4×C5 (order 80) · Inn(G) = C1 · Out(G) = C22×C4×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽11)×AGL(1,ℤ/16)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC176
abelianC11×C16

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC8C2×C11
socle, prime radicalC2×C11C8
LCSG ▷ C1
UCSC1 ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 176

A minimal faithful permutation representation — 1 generator on 27 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

center ofC11×M64, C11×#504, C11×C3⋊C32, …
derived subgroup ofD352, #10622087, Dic176, …
Frattini subgroup ofDic9680, D19360, Dic13552, …
Fitting subgroup ofC11⋊C32, C16×D11, #66862, …
maximal subgroup ofC11⋊C32, C11×C32, C16×D11, …
maximal normal subgroup ofC11⋊C32, C11×C32, C16×D11, …
maximal quotient ofC11×C32, C2×C11×C16, C11×C3⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 2);