Dic₈₀₉₁ #44569023

order	32364
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₉₁
unambiguous semidirect product	C₉⋊Dic₈₉₉
unambiguous semidirect product	C₃₁⋊Dic₂₆₁

Splittings

derived, Hall	C₉×C₂₉×C₃₁ ⋊ C₄
Hall	C₂₉×C₃₁ ⋊ Dic₉
Hall	C₉×C₃₁ ⋊ Dic₂₉
Hall	C₉×C₂₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₆₁
Sylow	C₂₉ ⋊ Dic₂₇₉
Sylow	C₉ ⋊ Dic₈₉₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₀₉₁
commutator, hypocenter	C₉×C₂₉×C₃₁	C₄
Frattini	C₂×C₃	D₂₆₉₇
Fitting, cosocle	C₂×C₉×C₂₉×C₃₁	C₂
socle, prime radical	C₂×C₃×C₂₉×C₃₁	D₃
LCS	G ▷ C₉×C₂₉×C₃₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 32364

A minimal faithful permutation representation — 2 generators on 73 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2,3,29,31,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^28,d^a=d^30,f^a=f^2

Conjugacy classes & character table

8094 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₁₈₂
maximal normal subgroup of	Dic₁₆₁₈₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32364, 9);

Dic8091 #44569023