D₁₁₀₄₉ #36348404

order	22098
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₀₄₉
unambiguous semidirect product	C₃⋊D₃₆₈₃
unambiguous semidirect product	C₂₉⋊D₃₈₁
unambiguous semidirect product	C₁₂₇⋊D₈₇

Splittings

Fitting, socle, derived, Hall	C₃×C₂₉×C₁₂₇ ⋊ C₂
Hall	C₂₉×C₁₂₇ ⋊ D₃
Hall	C₃×C₁₂₇ ⋊ D₂₉
Sylow	C₁₂₇ ⋊ D₈₇
Hall	C₃×C₂₉ ⋊ D₁₂₇
Sylow	C₂₉ ⋊ D₃₈₁
Sylow	C₃ ⋊ D₃₆₈₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₂₉×C₁₂₇	C₂
LCS	G ▷ C₃×C₂₉×C₁₂₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 11049

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

Polycyclic presentation:

2,3,29,127|b^a=b^2,c^a=c^28,d^a=d^126

Conjugacy classes & character table

5526 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22098, 9);

D11049 #36348404