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C11⋊(ΓL(1,𝔽8)⋊D61) #43245342

order28182
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC11⋊(ΓL(1,𝔽8)⋊D61)
unambiguous semidirect productΓL(1,𝔽8)⋊D671

Splittings

Fitting, socle, derived, HallC7×C11×C61 ⋊ C2×C3
HallC11×ΓL(1,𝔽8)×C61 ⋊ C2
HallD4697 ⋊ C3
HallΓL(1,𝔽8)×C61 ⋊ D11
HallC11×C61 ⋊ AGL(1,𝔽7)
HallC7×C61 ⋊ C3×D11
HallC11×ΓL(1,𝔽8) ⋊ D61
HallC7×C11 ⋊ C3×D61
SylowC61 ⋊ C11⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D671
SylowC11 ⋊ ΓL(1,𝔽8)⋊D61
SylowC7 ⋊ C3×D671

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C61C2×C3
LCSG ▷ C7×C11×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 4697

A minimal faithful permutation representation — 2 generators on 79 points:

(2,5,6,3,4,7)(9,10)(11,12)(13,14)(15,16)(17,18)(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);(1,2,4,7,6,5,3)(8,9,11,13,15,17,18,16,14,12,10)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21)

Polycyclic presentation:

2,3,7,11,61|c^a=c^6,d^a=d^10,e^a=e^60,c^b=c^2

Conjugacy classes & character table

1682 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28182, 46);