D₇₈₄₃ #30627976

order	15686
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₈₄₃
unambiguous semidirect product	C₁₁⋊D₇₁₃
unambiguous semidirect product	C₂₃⋊D₃₄₁
unambiguous semidirect product	C₃₁⋊D₂₅₃

Splittings

Fitting, socle, derived, Hall	C₁₁×C₂₃×C₃₁ ⋊ C₂
Hall	C₂₃×C₃₁ ⋊ D₁₁
Hall	C₁₁×C₃₁ ⋊ D₂₃
Hall	C₁₁×C₂₃ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₂₅₃
Sylow	C₂₃ ⋊ D₃₄₁
Sylow	C₁₁ ⋊ D₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₂₃×C₃₁	C₂
LCS	G ▷ C₁₁×C₂₃×C₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 7843

A minimal faithful permutation representation — 2 generators on 65 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,11,23,31|b^a=b^10,c^a=c^22,d^a=d^30

Conjugacy classes & character table

3923 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₅₃, D₂₃×D₃₄₁, D₁₁×D₇₁₃, …
maximal normal subgroup of	D₃₁×D₂₅₃, D₂₃×D₃₄₁, D₁₁×D₇₁₃, …
maximal quotient of	Dic₇₈₄₃, C₂×D₇₈₄₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15686, 9);

D7843 #30627976