D₈₅₆₉ #30686204

order	17138
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₅₆₉
unambiguous semidirect product	C₁₁⋊D₇₇₉
unambiguous semidirect product	C₁₉⋊D₄₅₁
unambiguous semidirect product	C₄₁⋊D₂₀₉

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₉×C₄₁ ⋊ C₂
Hall	C₁₉×C₄₁ ⋊ D₁₁
Hall	C₁₁×C₄₁ ⋊ D₁₉
Hall	C₁₁×C₁₉ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₂₀₉
Sylow	C₁₉ ⋊ D₄₅₁
Sylow	C₁₁ ⋊ D₇₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₉×C₄₁	C₂
LCS	G ▷ C₁₁×C₁₉×C₄₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 8569

A minimal faithful permutation representation — 2 generators on 71 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,11,19,41|b^a=b^10,c^a=c^18,d^a=d^40

Conjugacy classes & character table

4286 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₁×D₂₀₉, D₁₉×D₄₅₁, D₁₁×D₇₇₉, …
maximal normal subgroup of	D₄₁×D₂₀₉, D₁₉×D₄₅₁, D₁₁×D₇₇₉, …
maximal quotient of	Dic₈₅₆₉, C₂×D₈₅₆₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17138, 8);

D8569 #30686204