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C7×PSU(3,𝔽2)⋊C3 #13094887

order1512
isoclinismisoclinic class of PSU(3,𝔽2)⋊C3
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×PSU(3,𝔽2)⋊C3
unambiguous semidirect productPSU(3,𝔽2)⋊(C3×C7)

Splittings

Fitting, socleC32×C7 ⋊ SL(2,𝔽3)
derivedPSU(3,𝔽2) ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC7PSU(3,𝔽2)⋊C3
commutator, cosocle, hypocenterPSU(3,𝔽2)C3×C7
FrattiniC1G
Fitting, socleC32×C7SL(2,𝔽3)
LCSG ▷ PSU(3,𝔽2)
UCSC1 ◁ C7

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC2×ΓL(1,𝔽8)×AGL(2,𝔽3), C2×#72224672, C2×#72224678, …
maximal subgroup ofΓL(1,𝔽8)×PSU(3,𝔽2)⋊C3, D7×PSU(3,𝔽2)⋊C3, C7×AGL(2,𝔽3), …
maximal normal subgroup ofΓL(1,𝔽8)×PSU(3,𝔽2)⋊C3, D7×PSU(3,𝔽2)⋊C3, C7×AGL(2,𝔽3), …
maximal quotient ofC2×C7×PSU(3,𝔽2)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 784);