Dic₃₂₄₅ #27188587

order	12980
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₂₄₅
unambiguous semidirect product	C₁₁⋊Dic₂₉₅
unambiguous semidirect product	C₅₉⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₅₉ ⋊ C₄
Hall	C₁₁×C₅₉ ⋊ Dic₅
Hall	C₅×C₅₉ ⋊ Dic₁₁
Sylow	C₅₉ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₅₉
Sylow	C₁₁ ⋊ Dic₂₉₅
Sylow	C₅ ⋊ Dic₆₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₂₄₅
commutator, hypocenter	C₅×C₁₁×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₅₉	C₂
LCS	G ▷ C₅×C₁₁×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 12980

A minimal faithful permutation representation — 2 generators on 79 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);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,5,11,59,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^58

Conjugacy classes & character table

3248 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₄₉₀, Dic₁₆₂₂₅
maximal normal subgroup of	Dic₆₄₉₀
maximal quotient of	Dic₁₆₂₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12980, 9);

Dic3245 #27188587