Dic₂₉₁₅ #26042909

order	11660
automorphism	Aut(G) order 12126400
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₉₁₅
unambiguous semidirect product	C₁₁⋊Dic₂₆₅

Splittings

derived, Hall	C₅×C₁₁×C₅₃ ⋊ C₄
Hall	C₁₁×C₅₃ ⋊ Dic₅
Hall	C₅×C₅₃ ⋊ Dic₁₁
Hall	C₅×C₁₁ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₅₅
Sylow	C₁₁ ⋊ Dic₂₆₅
Sylow	C₅ ⋊ Dic₅₈₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₉₁₅
commutator, hypocenter	C₅×C₁₁×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₅₃	C₂
LCS	G ▷ C₅×C₁₁×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 11660

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)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2,5,11,53,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^52

Conjugacy classes & character table

2918 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₉₁₅, Dic₈₇₄₅, Dic₅₈₃₀, …
maximal normal subgroup of	C₃×Dic₂₉₁₅, Dic₅₈₃₀
maximal quotient of	C₃×Dic₂₉₁₅, Dic₈₇₄₅, Dic₁₄₅₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11660, 9);

Dic2915 #26042909