Dic₅₀₁₅ #34095398

order	20060
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₀₁₅
unambiguous semidirect product	C₅₉⋊Dic₈₅

Splittings

derived, Hall	C₅×C₁₇×C₅₉ ⋊ C₄
Hall	C₁₇×C₅₉ ⋊ Dic₅
Hall	C₅×C₅₉ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₂₉₅
Sylow	C₅ ⋊ Dic₁₀₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₀₁₅
commutator, hypocenter	C₅×C₁₇×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₅₉	C₂
LCS	G ▷ C₅×C₁₇×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 20060

A minimal faithful permutation representation — 2 generators on 85 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);(1,3,2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(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:

2,5,17,59,2|a^2=e,b^a=b^4,c^a=c^16,d^a=d^58

Conjugacy classes & character table

5018 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₃₀, Dic₁₅₀₄₅
maximal normal subgroup of	Dic₁₀₀₃₀
maximal quotient of	Dic₁₅₀₄₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20060, 8);

Dic5015 #34095398