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Cayley diagram

C11×#157 #87685

order396
isoclinismisoclinic class of #157
automorphismAut(G) = C2×C5×AΓL(1,𝔽9) (order 1440) · Inn(G) = #157 · Out(G) = C23×C5
homologyAbelianization (H1) = C4×C11 · Schur multiplier (H2) = C3

Aliases

direct productC11×#157

Splittings

Fitting, socle, HallC32×C11 ⋊ C4
derived, SylowC32 ⋊ C4×C11

Subgroups

subgroupG/H
center, hypercenterC11#157
commutator, hypocenterC32C4×C11
FrattiniC1G
Fitting, socleC32×C11C4
prime radicalC11×C3⋊D3C2
cosocleC3⋊D3C2×C11
LCSG ▷ C32
UCSC1 ◁ C11

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 66

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13097490, PSU(3,𝔽2)⋊D11, AGL(1,𝔽9)⋊D11, …
maximal subgroup ofC11×AGL(1,𝔽9), C11⋊AGL(1,𝔽9), D11×#157, …
maximal normal subgroup ofC11×AGL(1,𝔽9), C11⋊AGL(1,𝔽9), D11×#157, …
maximal quotient ofC11×#622, C2×C11×#157, C11×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(396, 17);