D₁₂₂₀₁ #37567367

order	24402
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₀₁
unambiguous semidirect product	C₃⋊D₄₀₆₇
unambiguous semidirect product	C₄₉⋊D₂₄₉
unambiguous semidirect product	C₈₃⋊D₁₄₇

Splittings

Fitting, derived, Hall	C₃×C₄₉×C₈₃ ⋊ C₂
Hall	C₄₉×C₈₃ ⋊ D₃
Hall	C₃×C₈₃ ⋊ D₄₉
Hall	C₃×C₄₉ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₁₄₇
Sylow	C₄₉ ⋊ D₂₄₉
Sylow	C₃ ⋊ D₄₀₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄₉×C₈₃	C₂
Frattini	C₇	D₁₇₄₃
socle	C₃×C₇×C₈₃	D₇
LCS	G ▷ C₃×C₄₉×C₈₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 12201

A minimal faithful permutation representation — 2 generators on 135 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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,3)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)

Polycyclic presentation:

2,3,7,83,7|c^7=e^6,b^a=b^2,c^a=c^6e,d^a=d^82,e^a=e^6

Conjugacy classes & character table

6102 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24402, 9);

D12201 #37567367