Dic₃₉₀₅ #30626781

order	15620
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₉₀₅
unambiguous semidirect product	C₁₁⋊Dic₃₅₅
unambiguous semidirect product	C₇₁⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₇₁ ⋊ C₄
Hall	C₁₁×C₇₁ ⋊ Dic₅
Hall	C₅×C₇₁ ⋊ Dic₁₁
Sylow	C₇₁ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₇₁
Sylow	C₁₁ ⋊ Dic₃₅₅
Sylow	C₅ ⋊ Dic₇₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₉₀₅
commutator, hypocenter	C₅×C₁₁×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₇₁	C₂
LCS	G ▷ C₅×C₁₁×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 15620

A minimal faithful permutation representation — 2 generators on 91 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,5,11,71,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^70

Conjugacy classes & character table

3908 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₈₁₀
maximal normal subgroup of	Dic₇₈₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15620, 14);

Dic3905 #30626781