Dic₄₁₆₅ #30659376

order	16660
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₁₆₅
unambiguous semidirect product	C₄₉⋊Dic₈₅

Splittings

derived, Hall	C₅×C₁₇×C₄₉ ⋊ C₄
Hall	C₁₇×C₄₉ ⋊ Dic₅
Hall	C₅×C₄₉ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₂₄₅
Sylow	C₅ ⋊ Dic₈₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₁₆₅
commutator, hypocenter	C₅×C₁₇×C₄₉	C₄
Frattini	C₂×C₇	D₅₉₅
Fitting, cosocle	C₂×C₅×C₁₇×C₄₉	C₂
socle, prime radical	C₂×C₅×C₇×C₁₇	D₇
LCS	G ▷ C₅×C₁₇×C₄₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 16660

A minimal faithful permutation representation — 2 generators on 75 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);(1,3,2,4)(6,9)(7,8)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2,5,7,17,2,7|a^2=e,c^7=f^6,b^a=b^4,c^a=c^6f,d^a=d^16,f^a=f^6

Conjugacy classes & character table

4168 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₃₃₀
maximal normal subgroup of	Dic₈₃₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16660, 8);

Dic4165 #30659376