D₁₀₉₇₁ #36346758

order	21942
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₉₇₁
unambiguous semidirect product	C₉⋊D₁₂₁₉
unambiguous semidirect product	C₂₃⋊D₄₇₇
unambiguous semidirect product	C₅₃⋊D₂₀₇

Splittings

Fitting, derived, Hall	C₉×C₂₃×C₅₃ ⋊ C₂
Hall	C₂₃×C₅₃ ⋊ D₉
Hall	C₉×C₅₃ ⋊ D₂₃
Hall	C₉×C₂₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₀₇
Sylow	C₂₃ ⋊ D₄₇₇
Sylow	C₉ ⋊ D₁₂₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₃×C₅₃	C₂
Frattini	C₃	D₃₆₅₇
socle	C₃×C₂₃×C₅₃	D₃
LCS	G ▷ C₉×C₂₃×C₅₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 10971

A minimal faithful permutation representation — 2 generators on 85 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);(2,9)(3,8)(4,7)(5,6)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,3,23,53,3|b^3=e^2,b^a=b^2e,c^a=c^22,d^a=d^52,e^a=e^2

Conjugacy classes & character table

5487 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21942, 8);

D10971 #36346758