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Cayley diagram

C32×#157 #66469

order324
isoclinismisoclinic class of #157
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C32

Aliases

direct productC32×#157

Splittings

Fitting, socle, SylowC34 ⋊ C4
derivedC32 ⋊ C32×C4

Subgroups

subgroupG/H
center, hypercenterC32#157
commutator, hypocenterC32C32×C4
FrattiniC1G
Fitting, socleC34C4
prime radicalC32×C3⋊D3C2
cosocleC3⋊D3C2×C32
LCSG ▷ C32
UCSC1 ◁ C32

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 12 points:

(4,5,6)(7,8)(9,10,12,11);(1,2,3)(7,9,12)(8,11,10)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13049796, #13049797, #13049798, …
maximal subgroup ofC32×AGL(1,𝔽9), C3×C3⋊AGL(1,𝔽9), C3⋊(C3⋊AGL(1,𝔽9)), …
maximal normal subgroup ofC32×AGL(1,𝔽9), C3×C3⋊AGL(1,𝔽9), C3⋊(C3⋊AGL(1,𝔽9)), …
maximal quotient ofC32×#622, C2×C32×#157, C3×C9×#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 161);