Finite Group Databasebeta

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Cayley diagram

C22×Dic11 #4344

order176
isoclinismisoclinic class of D11
automorphismAut(G) = AGL(1,𝔽11)×#5934 (order 21120) · Inn(G) = D11 · Out(G) = C5×#5934
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×Dic11

Splittings

derived, SylowC11 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23D11
commutator, hypocenterC11C22×C4
FrattiniC2C22×D11
Fitting, socle, prime radicalC23×C11C2
cosocleC2×C11C23
LCSG ▷ C11
UCSC1 ◁ C23

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 176

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2)(3,4)(5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19);(3,4)(9,10,12,14,16,18,19,17,15,13,11);(1,2)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11⋊#95, #66930, #66931, …
maximal normal subgroup ofC11⋊#95, #66930, #66931, …
maximal quotient ofC22×C11⋊C8, C2×#4318, C2×C4×Dic11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 35);