Dic₆₀₃₉ #37563451

order	24156
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₃₉
unambiguous semidirect product	C₉⋊Dic₆₇₁
unambiguous semidirect product	C₁₁⋊Dic₅₄₉

Splittings

derived, Hall	C₉×C₁₁×C₆₁ ⋊ C₄
Hall	C₁₁×C₆₁ ⋊ Dic₉
Hall	C₉×C₆₁ ⋊ Dic₁₁
Hall	C₉×C₁₁ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₉₉
Sylow	C₁₁ ⋊ Dic₅₄₉
Sylow	C₉ ⋊ Dic₆₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₀₃₉
commutator, hypocenter	C₉×C₁₁×C₆₁	C₄
Frattini	C₂×C₃	D₂₀₁₃
Fitting, cosocle	C₂×C₉×C₁₁×C₆₁	C₂
socle, prime radical	C₂×C₃×C₁₁×C₆₁	D₃
LCS	G ▷ C₉×C₁₁×C₆₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 24156

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)(6,13)(7,12)(8,11)(9,10)(15,24)(16,23)(17,22)(18,21)(19,20)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,3,11,61,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^60,f^a=f^2

Conjugacy classes & character table

6042 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₀₇₈
maximal normal subgroup of	Dic₁₂₀₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24156, 9);

Dic6039 #37563451