Dic₆₀₀₃ #37562146

order	24012
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₀₃
unambiguous semidirect product	C₉⋊Dic₆₆₇
unambiguous semidirect product	C₂₃⋊Dic₂₆₁

Splittings

derived, Hall	C₉×C₂₃×C₂₉ ⋊ C₄
Hall	C₂₃×C₂₉ ⋊ Dic₉
Hall	C₉×C₂₉ ⋊ Dic₂₃
Hall	C₉×C₂₃ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₂₀₇
Sylow	C₂₃ ⋊ Dic₂₆₁
Sylow	C₉ ⋊ Dic₆₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₀₀₃
commutator, hypocenter	C₉×C₂₃×C₂₉	C₄
Frattini	C₂×C₃	D₂₀₀₁
Fitting, cosocle	C₂×C₉×C₂₃×C₂₉	C₂
socle, prime radical	C₂×C₃×C₂₃×C₂₉	D₃
LCS	G ▷ C₉×C₂₃×C₂₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 24012

A minimal faithful permutation representation — 2 generators on 65 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,3,23,29,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^22,d^a=d^28,f^a=f^2

Conjugacy classes & character table

6006 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₀₀₆
maximal normal subgroup of	Dic₁₂₀₀₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24012, 8);

Dic6003 #37562146