D₁₀₇₆₉ #36340586

order	21538
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₇₆₉
unambiguous semidirect product	C₈₉⋊D₁₂₁
unambiguous semidirect product	C₁₂₁⋊D₈₉

Splittings

Fitting, derived, Hall	C₈₉×C₁₂₁ ⋊ C₂
Sylow	C₁₂₁ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₈₉×C₁₂₁	C₂
Frattini	C₁₁	D₉₇₉
socle	C₁₁×C₈₉	D₁₁
LCS	G ▷ C₈₉×C₁₂₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 210
transitive degree	μ_t(G) = 10769

A minimal faithful permutation representation — 2 generators on 210 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)(122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210);(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,200)(134,199)(135,198)(136,197)(137,196)(138,195)(139,194)(140,193)(141,192)(142,191)(143,190)(144,189)(145,188)(146,187)(147,186)(148,185)(149,184)(150,183)(151,182)(152,181)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)

Polycyclic presentation:

2,11,89,11|b^11=d^10,b^a=b^10d,c^a=c^88,d^a=d^10

Conjugacy classes & character table

5386 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₂₃₀₇
maximal quotient of	D₃₂₃₀₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21538, 6);

D10769 #36340586