D₁₀₁₉₉ #35205209

order	20398
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₁₉₉
unambiguous semidirect product	C₇⋊D₁₄₅₇
unambiguous semidirect product	C₃₁⋊D₃₂₉
unambiguous semidirect product	C₄₇⋊D₂₁₇

Splittings

Fitting, socle, derived, Hall	C₇×C₃₁×C₄₇ ⋊ C₂
Hall	C₃₁×C₄₇ ⋊ D₇
Hall	C₇×C₄₇ ⋊ D₃₁
Hall	C₇×C₃₁ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₂₁₇
Sylow	C₃₁ ⋊ D₃₂₉
Sylow	C₇ ⋊ D₁₄₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₁×C₄₇	C₂
LCS	G ▷ C₇×C₃₁×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 10199

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);(2,7)(3,6)(4,5)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2,7,31,47|b^a=b^6,c^a=c^30,d^a=d^46

Conjugacy classes & character table

5101 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20398, 8);

D10199 #35205209