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C2×C5×ΓL(1,𝔽8)⋊D89 #49153152

order37380
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×ΓL(1,𝔽8)⋊D89
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C5×D89)
unambiguous semidirect product(C2×C89)⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product(C2×D623)⋊(C3×C5)
unambiguous semidirect product(C2×C5×D623)⋊C3

Splittings

Fitting, socleC2×C5×C7×C89 ⋊ C2×C3
derived, HallC7×C89 ⋊ C22×C3×C5
HallC2×C5×D623 ⋊ C3
HallC5×ΓL(1,𝔽8)×C89 ⋊ C22
HallC5×C7×C89 ⋊ C22×C3
HallC2×D623 ⋊ C3×C5
HallΓL(1,𝔽8)×C89 ⋊ C22×C5
HallC5×C89 ⋊ C2×AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ C2×D89
SylowC89 ⋊ C2×C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3×D89
HallΓL(1,𝔽8) ⋊ C2×C5×D89
SylowC7 ⋊ C2×C3×C5×D89

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)⋊D89
commutator, cosocle, hypocenterC7×C89C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C89C2×C3
LCSG ▷ C7×C89
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 103
transitive degreeμt(G) = 6230

A minimal faithful permutation representation — 2 generators on 103 points:

(3,4,5,6,7)(9,12,13,10,11,14)(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)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103);(1,2)(8,9,11,14,13,12,10)(15,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17)

Conjugacy classes & character table

2270 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37380, 46);