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ΓL(1,𝔽8)×C23×A5 #44394686

order28980
automorphism?
homologyAbelianization (H1) = C3×C23 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×C23×A5
unambiguous semidirect productC7⋊(C3×C23×A5)
unambiguous semidirect product(C7×C23)⋊(C3×A5)
unambiguous semidirect product(C7×A5)⋊(C3×C23)
unambiguous semidirect product(C7×C23×A5)⋊C3

Splittings

Fitting, HallC7×C23 ⋊ C3×A5
socleC7×C23×A5 ⋊ C3
derivedC7×A5 ⋊ C3×C23
SylowC7 ⋊ C3×C23×A5

Subgroups

subgroupG/H
center, hypercenterC23ΓL(1,𝔽8)×A5
commutator, hypocenterC7×A5C3×C23
FrattiniC1G
FittingC7×C23C3×A5
socleC7×C23×A5C3
cosocleC7C3×C23×A5
LCSG ▷ C7×A5
UCSC1 ◁ C23

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 805

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

575 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28980, 355);