Finite Group Databasebeta

← Prev (#27189150) Next (#27189152) →

AGL(1,𝔽5)×ΓL(1,𝔽8)×C31 #27189151

order13020
automorphism?
homologyAbelianization (H1) = C3×C4×C31 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×ΓL(1,𝔽8)×C31
unambiguous semidirect product(C7×AGL(1,𝔽5))⋊(C3×C31)

Splittings

Fitting, socle, HallC5×C7×C31 ⋊ C3×C4
derived, HallC5×C7 ⋊ C3×C4×C31
HallC7×AGL(1,𝔽5)×C31 ⋊ C3
HallC5×ΓL(1,𝔽8)×C31 ⋊ C4
HallC7×C31 ⋊ C3×AGL(1,𝔽5)
HallC5×C31 ⋊ C4×ΓL(1,𝔽8)
HallC7×AGL(1,𝔽5) ⋊ C3×C31
HallC5×ΓL(1,𝔽8) ⋊ C4×C31
SylowC7 ⋊ C3×AGL(1,𝔽5)×C31
SylowC5 ⋊ C4×ΓL(1,𝔽8)×C31

Subgroups

subgroupG/H
center, hypercenterC31AGL(1,𝔽5)×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C3×C4×C31
FrattiniC1G
Fitting, socleC5×C7×C31C3×C4
prime radicalD5×ΓL(1,𝔽8)×C31C2
cosocleC7×D5C2×C3×C31
LCSG ▷ C5×C7
UCSC1 ◁ C31

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 1085

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3,4)(5,7,6)(9,12,11,10)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(1,2,3,5,4,6,7)(8,9,10,12,11)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 118);