Dic₈₀₄₁ #44565485

order	32164
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₄₁
unambiguous semidirect product	C₁₁⋊Dic₇₃₁
unambiguous semidirect product	C₄₃⋊Dic₁₈₇

Splittings

derived, Hall	C₁₁×C₁₇×C₄₃ ⋊ C₄
Hall	C₁₇×C₄₃ ⋊ Dic₁₁
Hall	C₁₁×C₄₃ ⋊ Dic₁₇
Hall	C₁₁×C₁₇ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₈₇
Sylow	C₁₇ ⋊ Dic₄₇₃
Sylow	C₁₁ ⋊ Dic₇₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₄₁
commutator, hypocenter	C₁₁×C₁₇×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₇×C₄₃	C₂
LCS	G ▷ C₁₁×C₁₇×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 32164

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)(6,15)(7,14)(8,13)(9,12)(10,11)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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:

2,11,17,43,2|a^2=e,b^a=b^10,c^a=c^16,d^a=d^42

Conjugacy classes & character table

8044 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₀₈₂
maximal normal subgroup of	Dic₁₆₀₈₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32164, 8);

Dic8041 #44565485