D₁₀₆₁₉ #35234802

order	21238
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₆₁₉
unambiguous semidirect product	C₇⋊D₁₅₁₇
unambiguous semidirect product	C₃₇⋊D₂₈₇
unambiguous semidirect product	C₄₁⋊D₂₅₉

Splittings

Fitting, socle, derived, Hall	C₇×C₃₇×C₄₁ ⋊ C₂
Hall	C₃₇×C₄₁ ⋊ D₇
Hall	C₇×C₄₁ ⋊ D₃₇
Hall	C₇×C₃₇ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₂₅₉
Sylow	C₃₇ ⋊ D₂₈₇
Sylow	C₇ ⋊ D₁₅₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₇×C₄₁	C₂
LCS	G ▷ C₇×C₃₇×C₄₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 10619

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)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Polycyclic presentation:

2,7,37,41|b^a=b^6,c^a=c^36,d^a=d^40

Conjugacy classes & character table

5311 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21238, 8);

D10619 #35234802