D₈₂₁₅ #30656133

order	16430
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₂₁₅
unambiguous semidirect product	C₅⋊D₁₆₄₃
unambiguous semidirect product	C₃₁⋊D₂₆₅
unambiguous semidirect product	C₅₃⋊D₁₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₁×C₅₃ ⋊ C₂
Hall	C₃₁×C₅₃ ⋊ D₅
Hall	C₅×C₅₃ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₂₆₅
Sylow	C₅ ⋊ D₁₆₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₁×C₅₃	C₂
LCS	G ▷ C₅×C₃₁×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 8215

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);(2,5)(3,4)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(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:

2,5,31,53|b^a=b^4,c^a=c^30,d^a=d^52

Conjugacy classes & character table

4109 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₅₅, D₃₁×D₂₆₅, D₅×D₁₆₄₃, …
maximal normal subgroup of	D₅₃×D₁₅₅, D₃₁×D₂₆₅, D₅×D₁₆₄₃, …
maximal quotient of	Dic₈₂₁₅, C₂×D₈₂₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16430, 9);

D8215 #30656133