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C5×C7⋊H(3,𝔽3) #11745705

order945
isoclinismisoclinic class of #4427
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C32

Aliases

direct productC5×C7⋊H(3,𝔽3)
unambiguous semidirect productC7⋊(C5×H(3,𝔽3))
unambiguous semidirect productC32⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C5×C7)⋊H(3,𝔽3)
unambiguous semidirect product(C32×C5)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C3×ΓL(1,𝔽8))⋊(C3×C5)
unambiguous semidirect product(C3×C5×ΓL(1,𝔽8))⋊C3

Splittings

FittingC32×C5×C7 ⋊ C3
HallC5×C7 ⋊ H(3,𝔽3)
SylowC7 ⋊ C5×H(3,𝔽3)

Subgroups

subgroupG/H
centerC3×C5C3×ΓL(1,𝔽8)
commutator, cosocleC3×C7C32×C5
FrattiniC3C3×C5×ΓL(1,𝔽8)
FittingC32×C5×C7C3
socleC3×C5×C7C32
hypercenterC32×C5ΓL(1,𝔽8)
hypocenterC7C5×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3×C5 ◁ C32×C5

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 315

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4,5)(6,7,8,10,9,11,12)(14,17,18)(16,20,21);(7,8,9)(10,12,11)(13,14,16)(15,18,20)(17,21,19)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×D7⋊H(3,𝔽3), D5×C7⋊H(3,𝔽3), C5⋊(D7⋊H(3,𝔽3)), …
maximal normal subgroup ofC5×D7⋊H(3,𝔽3), D5×C7⋊H(3,𝔽3), C5⋊(D7⋊H(3,𝔽3)), …
maximal quotient ofC2×C5×C7⋊H(3,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(945, 8);