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C2×C5×ΓL(1,𝔽8)×D31 #27189199

order13020
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×ΓL(1,𝔽8)×D31

Splittings

Fitting, socleC2×C5×C7×C31 ⋊ C2×C3
derived, HallC7×C31 ⋊ C22×C3×C5
HallC2×C5×C7×D31 ⋊ C3
HallC5×ΓL(1,𝔽8)×C31 ⋊ C22
HallC5×C7×C31 ⋊ C22×C3
HallC2×C7×D31 ⋊ C3×C5
HallΓL(1,𝔽8)×C31 ⋊ C22×C5
HallC5×C31 ⋊ C22×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C3×D31
SylowC31 ⋊ C22×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×C5×D31

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×D31
commutator, cosocle, hypocenterC7×C31C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C31C2×C3
LCSG ▷ C7×C31
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 2170

A minimal faithful permutation representation — 2 generators on 45 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(33,34,35)(36,38,37)(39,40,41,42,43);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(32,33,34,36,35,37,38)(44,45)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 166);