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C2×C7×ΓL(1,𝔽32) #14524239

order2170
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphism?
homologyAbelianization (H1) = C2×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC2×C7×ΓL(1,𝔽32)
unambiguous semidirect product(C2×C31)⋊(C5×C7)
unambiguous semidirect product(C2×C7×C31)⋊C5

Splittings

Fitting, socle, HallC2×C7×C31 ⋊ C5
derived, SylowC31 ⋊ C2×C5×C7
HallC7×C31 ⋊ C2×C5
HallC2×C31 ⋊ C5×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽32)
commutator, cosocle, hypocenterC31C2×C5×C7
FrattiniC1G
Fitting, socleC2×C7×C31C5
LCSG ▷ C31
UCSC1 ◁ C2×C7

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 434

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,4,5,6,7,8,9)(11,12,14,17,13)(15,19,24,27,16)(18,22,23,21,20)(25,31,35,33,26)(28,32,30,34,29)(36,40,39,38,37);(10,11,13,16,17,21,27,33,14,18,23,29,24,30,35,39,12,15,20,26,22,28,34,38,19,25,32,37,31,36,40)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×C7×ΓL(1,𝔽32), C7×Dic31⋊C5, Dic7×ΓL(1,𝔽32), …
maximal normal subgroup ofC4×C7×ΓL(1,𝔽32), C7×Dic31⋊C5, Dic7×ΓL(1,𝔽32), …
maximal quotient ofC4×C7×ΓL(1,𝔽32), C22×C7×ΓL(1,𝔽32), C2×C7×C31⋊C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2170, 2);