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C5⋊(ΓL(1,𝔽8)⋊D139) #44415936

order29190
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(ΓL(1,𝔽8)⋊D139)
unambiguous semidirect productΓL(1,𝔽8)⋊D695

Splittings

Fitting, socle, derived, HallC5×C7×C139 ⋊ C2×C3
HallC5×ΓL(1,𝔽8)×C139 ⋊ C2
HallD4865 ⋊ C3
HallΓL(1,𝔽8)×C139 ⋊ D5
HallC7×C139 ⋊ C3×D5
HallC5×C139 ⋊ AGL(1,𝔽7)
SylowC139 ⋊ C5⋊AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ D139
HallC5×C7 ⋊ C3×D139
HallΓL(1,𝔽8) ⋊ D695
SylowC7 ⋊ C3×D695
SylowC5 ⋊ ΓL(1,𝔽8)⋊D139

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C139C2×C3
LCSG ▷ C5×C7×C139
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 151
transitive degreeμt(G) = 4865

A minimal faithful permutation representation — 2 generators on 151 points:

(2,3)(4,5)(7,10,11,8,9,12)(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)(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)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147)(148,149)(150,151);(1,2,4,5,3)(6,7,9,12,11,10,8)(13,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150,151,149,147,145,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15)

Polycyclic presentation:

2,3,5,7,139|c^a=c^4,d^a=d^6,e^a=e^138,d^b=d^2

Conjugacy classes & character table

1742 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29190, 46);