Dic₄₀₅₉ #30634065

order	16236
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₅₉
unambiguous semidirect product	C₉⋊Dic₄₅₁
unambiguous semidirect product	C₁₁⋊Dic₃₆₉

Splittings

derived, Hall	C₉×C₁₁×C₄₁ ⋊ C₄
Hall	C₁₁×C₄₁ ⋊ Dic₉
Hall	C₉×C₄₁ ⋊ Dic₁₁
Hall	C₉×C₁₁ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₉₉
Sylow	C₁₁ ⋊ Dic₃₆₉
Sylow	C₉ ⋊ Dic₄₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₀₅₉
commutator, hypocenter	C₉×C₁₁×C₄₁	C₄
Frattini	C₂×C₃	D₁₃₅₃
Fitting, cosocle	C₂×C₉×C₁₁×C₄₁	C₂
socle, prime radical	C₂×C₃×C₁₁×C₄₁	D₃
LCS	G ▷ C₉×C₁₁×C₄₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 16236

A minimal faithful permutation representation — 2 generators on 65 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,24)(16,23)(17,22)(18,21)(19,20)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2,3,11,41,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^40,f^a=f^2

Conjugacy classes & character table

4062 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₁₁₈, Dic₁₂₁₇₇
maximal normal subgroup of	Dic₈₁₁₈
maximal quotient of	Dic₁₂₁₇₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16236, 8);

Dic4059 #30634065