D₁₀₇₅₉ #36340245

order	21518
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₇₅₉
unambiguous semidirect product	C₇⋊D₁₅₃₇
unambiguous semidirect product	C₂₉⋊D₃₇₁
unambiguous semidirect product	C₅₃⋊D₂₀₃

Splittings

Fitting, socle, derived, Hall	C₇×C₂₉×C₅₃ ⋊ C₂
Hall	C₂₉×C₅₃ ⋊ D₇
Hall	C₇×C₅₃ ⋊ D₂₉
Hall	C₇×C₂₉ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₀₃
Sylow	C₂₉ ⋊ D₃₇₁
Sylow	C₇ ⋊ D₁₅₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₉×C₅₃	C₂
LCS	G ▷ C₇×C₂₉×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 10759

A minimal faithful permutation representation — 2 generators on 89 points:

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

Polycyclic presentation:

2,7,29,53|b^a=b^6,c^a=c^28,d^a=d^52

Conjugacy classes & character table

5381 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21518, 9);

D10759 #36340245