D₁₂₂₆₁ #37589227

order	24522
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₆₁
unambiguous semidirect product	C₃⋊D₄₀₈₇
unambiguous semidirect product	C₆₁⋊D₂₀₁
unambiguous semidirect product	C₆₇⋊D₁₈₃

Splittings

Fitting, socle, derived, Hall	C₃×C₆₁×C₆₇ ⋊ C₂
Hall	C₆₁×C₆₇ ⋊ D₃
Hall	C₃×C₆₇ ⋊ D₆₁
Hall	C₃×C₆₁ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₁₈₃
Sylow	C₆₁ ⋊ D₂₀₁
Sylow	C₃ ⋊ D₄₀₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₆₁×C₆₇	C₂
LCS	G ▷ C₃×C₆₁×C₆₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 12261

A minimal faithful permutation representation — 2 generators on 131 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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(2,3)(5,64)(6,63)(7,62)(8,61)(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)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)

Polycyclic presentation:

2,3,61,67|b^a=b^2,c^a=c^60,d^a=d^66

Conjugacy classes & character table

6132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24522, 12);

D12261 #37589227