Dic₄₉₅₉ #34090593

order	19836
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₉₅₉
unambiguous semidirect product	C₉⋊Dic₅₅₁
unambiguous semidirect product	C₁₉⋊Dic₂₆₁

Splittings

derived, Hall	C₉×C₁₉×C₂₉ ⋊ C₄
Hall	C₁₉×C₂₉ ⋊ Dic₉
Hall	C₉×C₂₉ ⋊ Dic₁₉
Hall	C₉×C₁₉ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₁₇₁
Sylow	C₁₉ ⋊ Dic₂₆₁
Sylow	C₉ ⋊ Dic₅₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₉₅₉
commutator, hypocenter	C₉×C₁₉×C₂₉	C₄
Frattini	C₂×C₃	D₁₆₅₃
Fitting, cosocle	C₂×C₉×C₁₉×C₂₉	C₂
socle, prime radical	C₂×C₃×C₁₉×C₂₉	D₃
LCS	G ▷ C₉×C₁₉×C₂₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 19836

A minimal faithful permutation representation — 2 generators on 61 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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:

2,3,19,29,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^18,d^a=d^28,f^a=f^2

Conjugacy classes & character table

4962 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₉₁₈, Dic₁₄₈₇₇
maximal normal subgroup of	Dic₉₉₁₈
maximal quotient of	Dic₁₄₈₇₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19836, 9);

Dic4959 #34090593