Dic₆₁₄₁ #37589556

order	24564
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₄₁
unambiguous semidirect product	C₃⋊Dic₂₀₄₇
unambiguous semidirect product	C₂₃⋊Dic₂₆₇

Splittings

derived, Hall	C₃×C₂₃×C₈₉ ⋊ C₄
Hall	C₂₃×C₈₉ ⋊ Dic₃
Hall	C₃×C₈₉ ⋊ Dic₂₃
Sylow	C₈₉ ⋊ Dic₆₉
Hall	C₃×C₂₃ ⋊ Dic₈₉
Sylow	C₂₃ ⋊ Dic₂₆₇
Sylow	C₃ ⋊ Dic₂₀₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁₄₁
commutator, hypocenter	C₃×C₂₃×C₈₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₈₉	C₂
LCS	G ▷ C₃×C₂₃×C₈₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 24564

A minimal faithful permutation representation — 2 generators on 119 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3,2,4)(6,7)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2,3,23,89,2|a^2=e,b^a=b^2,c^a=c^22,d^a=d^88

Conjugacy classes & character table

6144 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₂₈₂
maximal normal subgroup of	Dic₁₂₂₈₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24564, 8);

Dic6141 #37589556