D₁₂₁₈₃ #37567054

order	24366
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₁₈₃
unambiguous semidirect product	C₃⋊D₄₀₆₁
unambiguous semidirect product	C₃₁⋊D₃₉₃
unambiguous semidirect product	C₁₃₁⋊D₉₃

Splittings

Fitting, socle, derived, Hall	C₃×C₃₁×C₁₃₁ ⋊ C₂
Hall	C₃₁×C₁₃₁ ⋊ D₃
Hall	C₃×C₁₃₁ ⋊ D₃₁
Sylow	C₁₃₁ ⋊ D₉₃
Hall	C₃×C₃₁ ⋊ D₁₃₁
Sylow	C₃₁ ⋊ D₃₉₃
Sylow	C₃ ⋊ D₄₀₆₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₃₁×C₁₃₁	C₂
LCS	G ▷ C₃×C₃₁×C₁₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 12183

A minimal faithful permutation representation — 2 generators on 165 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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165);(2,3)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)

Polycyclic presentation:

2,3,31,131|b^a=b^2,c^a=c^30,d^a=d^130

Conjugacy classes & character table

6093 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24366, 9);

D12183 #37567054