Dic₄₄₉₅ #31822874

order	17980
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₄₉₅
unambiguous semidirect product	C₃₁⋊Dic₁₄₅

Splittings

derived, Hall	C₅×C₂₉×C₃₁ ⋊ C₄
Hall	C₂₉×C₃₁ ⋊ Dic₅
Hall	C₅×C₃₁ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₁₅₅
Sylow	C₅ ⋊ Dic₈₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₄₉₅
commutator, hypocenter	C₅×C₂₉×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₃₁	C₂
LCS	G ▷ C₅×C₂₉×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 17980

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)(6,9)(7,8)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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:

2,5,29,31,2|a^2=e,b^a=b^4,c^a=c^28,d^a=d^30

Conjugacy classes & character table

4498 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₉₉₀, Dic₁₃₄₈₅
maximal normal subgroup of	Dic₈₉₉₀
maximal quotient of	Dic₁₃₄₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17980, 9);

Dic4495 #31822874