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D7×ΓL(1,𝔽32) #14524248

order2170
isoclinismisoclinic class of D7×ΓL(1,𝔽32)
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productD7×ΓL(1,𝔽32)
unambiguous semidirect productC7⋊(C2×ΓL(1,𝔽32))
unambiguous semidirect product(D7×C31)⋊C5

Splittings

Fitting, socle, derived, HallC7×C31 ⋊ C2×C5
HallC7×ΓL(1,𝔽32) ⋊ C2
HallD7×C31 ⋊ C5
SylowC31 ⋊ C5×D7
SylowC7 ⋊ C2×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C31C2×C5
LCSG ▷ C7×C31
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 217

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,4,5,6,7)(9,10,12,15,11)(13,17,22,25,14)(16,20,21,19,18)(23,29,33,31,24)(26,30,28,32,27)(34,38,37,36,35);(2,7)(3,6)(4,5)(8,9,11,14,15,19,25,31,12,16,21,27,22,28,33,37,10,13,18,24,20,26,32,36,17,23,30,35,29,34,38)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D31⋊C5, C2×D7×ΓL(1,𝔽32), C5×D7×ΓL(1,𝔽32), …
maximal normal subgroup ofD7×D31⋊C5, C2×D7×ΓL(1,𝔽32), C5×D7×ΓL(1,𝔽32), …
maximal quotient ofDic7×ΓL(1,𝔽32), C2×D7×ΓL(1,𝔽32), D7×C31⋊C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2170, 11);