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C5⋊(ΓL(1,𝔽8)⋊D101) #35234606

order21210
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(ΓL(1,𝔽8)⋊D101)
unambiguous semidirect productΓL(1,𝔽8)⋊D505
unambiguous semidirect productC101⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect productD3535⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C101 ⋊ C2×C3
HallC5×ΓL(1,𝔽8)×C101 ⋊ C2
HallD3535 ⋊ C3
HallΓL(1,𝔽8)×C101 ⋊ D5
HallC7×C101 ⋊ C3×D5
HallC5×C101 ⋊ AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ D101
SylowC101 ⋊ C5⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D101
HallΓL(1,𝔽8) ⋊ D505
SylowC7 ⋊ C3×D505
SylowC5 ⋊ ΓL(1,𝔽8)⋊D101

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C101C2×C3
LCSG ▷ C5×C7×C101
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 113
transitive degreeμt(G) = 3535

A minimal faithful permutation representation — 2 generators on 113 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)(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)(103,106,107,104,105,108)(110,111)(112,113);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(102,103,105,108,107,106,104)(109,110,112,113,111)

Polycyclic presentation:

2,3,5,7,101|c^a=c^4,d^a=d^6,e^a=e^100,d^b=d^2

Conjugacy classes & character table

1267 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21210, 33);