Dic₆₀₆₁ #37564607

order	24244
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₆₁
unambiguous semidirect product	C₁₁⋊Dic₅₅₁
unambiguous semidirect product	C₁₉⋊Dic₃₁₉

Splittings

derived, Hall	C₁₁×C₁₉×C₂₉ ⋊ C₄
Hall	C₁₉×C₂₉ ⋊ Dic₁₁
Hall	C₁₁×C₂₉ ⋊ Dic₁₉
Hall	C₁₁×C₁₉ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₂₀₉
Sylow	C₁₉ ⋊ Dic₃₁₉
Sylow	C₁₁ ⋊ Dic₅₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₀₆₁
commutator, hypocenter	C₁₁×C₁₉×C₂₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₉×C₂₉	C₂
LCS	G ▷ C₁₁×C₁₉×C₂₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 24244

A minimal faithful permutation representation — 2 generators on 63 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,11,19,29,2|a^2=e,b^a=b^10,c^a=c^18,d^a=d^28

Conjugacy classes & character table

6064 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₁₂₂
maximal normal subgroup of	Dic₁₂₁₂₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24244, 8);

Dic6061 #37564607