Dic₃₉₆₅ #30629415

order	15860
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₉₆₅

Splittings

derived, Hall	C₅×C₁₃×C₆₁ ⋊ C₄
Hall	C₁₃×C₆₁ ⋊ Dic₅
Hall	C₅×C₆₁ ⋊ Dic₁₃
Hall	C₅×C₁₃ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₆₅
Sylow	C₁₃ ⋊ Dic₃₀₅
Sylow	C₅ ⋊ Dic₇₉₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₉₆₅
commutator, hypocenter	C₅×C₁₃×C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₆₁	C₂
LCS	G ▷ C₅×C₁₃×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 15860

A minimal faithful permutation representation — 2 generators on 83 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);(1,3,2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,5,13,61,2|a^2=e,b^a=b^4,c^a=c^12,d^a=d^60

Conjugacy classes & character table

3968 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₉₃₀
maximal normal subgroup of	Dic₇₉₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15860, 9);

Dic3965 #30629415