Dic₁₀₄₅ #16862017

order	4180
automorphism	Aut(G) order 1504800
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₀₄₅
unambiguous semidirect product	C₁₁⋊Dic₉₅
unambiguous semidirect product	C₁₉⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₁₉ ⋊ C₄
Hall	C₁₁×C₁₉ ⋊ Dic₅
Hall	C₅×C₁₉ ⋊ Dic₁₁
Hall	C₅×C₁₁ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₅₅
Sylow	C₁₁ ⋊ Dic₉₅
Sylow	C₅ ⋊ Dic₂₀₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₀₄₅
commutator, hypocenter	C₅×C₁₁×C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₁₉	C₂
LCS	G ▷ C₅×C₁₁×C₁₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 4180

A minimal faithful permutation representation — 2 generators on 39 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);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Polycyclic presentation:

2,5,11,19,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^18

Conjugacy classes & character table

1048 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₁₀₄₅, Dic₃₁₃₅, C₁₁⋊((C₁₉⋊C₃)⋊Dic₅), …
maximal normal subgroup of	C₃×Dic₁₀₄₅, C₁₁⋊((C₁₉⋊C₃)⋊Dic₅), C₁₉⋊((C₁₁⋊C₅)⋊Dic₅), …
maximal quotient of	C₃×Dic₁₀₄₅, Dic₃₁₃₅, Dic₅₂₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4180, 9);

Dic1045 #16862017