Dic₅₆₈₁ #36384341

order	22724
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₆₈₁
unambiguous semidirect product	C₁₉⋊Dic₂₉₉
unambiguous semidirect product	C₂₃⋊Dic₂₄₇

Splittings

derived, Hall	C₁₃×C₁₉×C₂₃ ⋊ C₄
Hall	C₁₉×C₂₃ ⋊ Dic₁₃
Hall	C₁₃×C₂₃ ⋊ Dic₁₉
Hall	C₁₃×C₁₉ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₂₄₇
Sylow	C₁₉ ⋊ Dic₂₉₉
Sylow	C₁₃ ⋊ Dic₄₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₆₈₁
commutator, hypocenter	C₁₃×C₁₉×C₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₁₉×C₂₃	C₂
LCS	G ▷ C₁₃×C₁₉×C₂₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 22724

A minimal faithful permutation representation — 2 generators on 59 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,13,19,23,2|a^2=e,b^a=b^12,c^a=c^18,d^a=d^22

Conjugacy classes & character table

5684 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₃₆₂
maximal normal subgroup of	Dic₁₁₃₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22724, 8);

Dic5681 #36384341