Dic₉₃₈₁ #49176171

order	37524
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₃₈₁
unambiguous semidirect product	C₃⋊Dic₃₁₂₇
unambiguous semidirect product	C₅₉⋊Dic₁₅₉

Splittings

derived, Hall	C₃×C₅₃×C₅₉ ⋊ C₄
Hall	C₅₃×C₅₉ ⋊ Dic₃
Hall	C₃×C₅₉ ⋊ Dic₅₃
Hall	C₃×C₅₃ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₅₉
Sylow	C₅₃ ⋊ Dic₁₇₇
Sylow	C₃ ⋊ Dic₃₁₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₃₈₁
commutator, hypocenter	C₃×C₅₃×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅₃×C₅₉	C₂
LCS	G ▷ C₃×C₅₃×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 37524

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

Polycyclic presentation:

2,3,53,59,2|a^2=e,b^a=b^2,c^a=c^52,d^a=d^58

Conjugacy classes & character table

9384 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37524, 8);

Dic9381 #49176171