Dic₃₆₈₅ #28413896

order	14740
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₆₈₅
unambiguous semidirect product	C₁₁⋊Dic₃₃₅
unambiguous semidirect product	C₆₇⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₆₇ ⋊ C₄
Hall	C₁₁×C₆₇ ⋊ Dic₅
Hall	C₅×C₆₇ ⋊ Dic₁₁
Sylow	C₆₇ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₆₇
Sylow	C₁₁ ⋊ Dic₃₃₅
Sylow	C₅ ⋊ Dic₇₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₆₈₅
commutator, hypocenter	C₅×C₁₁×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₆₇	C₂
LCS	G ▷ C₅×C₁₁×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 14740

A minimal faithful permutation representation — 2 generators on 87 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);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,5,11,67,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^66

Conjugacy classes & character table

3688 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₃₇₀
maximal normal subgroup of	Dic₇₃₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14740, 10);

Dic3685 #28413896