Dic₅₈₆₃ #37552982

order	23452
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₈₆₃
unambiguous semidirect product	C₁₁⋊Dic₅₃₃

Splittings

derived, Hall	C₁₁×C₁₃×C₄₁ ⋊ C₄
Hall	C₁₃×C₄₁ ⋊ Dic₁₁
Hall	C₁₁×C₄₁ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₄₅₁
Sylow	C₁₁ ⋊ Dic₅₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₈₆₃
commutator, hypocenter	C₁₁×C₁₃×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₄₁	C₂
LCS	G ▷ C₁₁×C₁₃×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 23452

A minimal faithful permutation representation — 2 generators on 69 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,11,13,41,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^40

Conjugacy classes & character table

5866 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₇₂₆
maximal normal subgroup of	Dic₁₁₇₂₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23452, 8);

Dic5863 #37552982