D₈₇₂₉ #31775438

order	17458
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₇₂₉
unambiguous semidirect product	C₇⋊D₁₂₄₇
unambiguous semidirect product	C₂₉⋊D₃₀₁
unambiguous semidirect product	C₄₃⋊D₂₀₃

Splittings

Fitting, socle, derived, Hall	C₇×C₂₉×C₄₃ ⋊ C₂
Hall	C₂₉×C₄₃ ⋊ D₇
Hall	C₇×C₄₃ ⋊ D₂₉
Hall	C₇×C₂₉ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₂₀₃
Sylow	C₂₉ ⋊ D₃₀₁
Sylow	C₇ ⋊ D₁₂₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₉×C₄₃	C₂
LCS	G ▷ C₇×C₂₉×C₄₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 8729

A minimal faithful permutation representation — 2 generators on 79 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);(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)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2,7,29,43|b^a=b^6,c^a=c^28,d^a=d^42

Conjugacy classes & character table

4366 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₃×D₂₀₃, D₂₉×D₃₀₁, D₇×D₁₂₄₇, …
maximal normal subgroup of	D₄₃×D₂₀₃, D₂₉×D₃₀₁, D₇×D₁₂₄₇, …
maximal quotient of	Dic₈₇₂₉, C₂×D₈₇₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17458, 16);

D8729 #31775438