D₁₃₂₆₁ #40997283

order	26522
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₂₆₁
unambiguous semidirect product	C₈₉⋊D₁₄₉
unambiguous semidirect product	C₁₄₉⋊D₈₉

Splittings

Fitting, socle, derived, Hall	C₈₉×C₁₄₉ ⋊ C₂
Sylow	C₁₄₉ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₁₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₈₉×C₁₄₉	C₂
LCS	G ▷ C₈₉×C₁₄₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 238
transitive degree	μ_t(G) = 13261

A minimal faithful permutation representation — 2 generators on 238 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)(90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238);(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(91,238)(92,237)(93,236)(94,235)(95,234)(96,233)(97,232)(98,231)(99,230)(100,229)(101,228)(102,227)(103,226)(104,225)(105,224)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,217)(113,216)(114,215)(115,214)(116,213)(117,212)(118,211)(119,210)(120,209)(121,208)(122,207)(123,206)(124,205)(125,204)(126,203)(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)(142,187)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)

Polycyclic presentation:

2,89,149|b^a=b^88,c^a=c^148

Conjugacy classes & character table

6632 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₂₆₁
maximal normal subgroup of	C₂×D₁₃₂₆₁
maximal quotient of	Dic₁₃₂₆₁, C₂×D₁₃₂₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26522, 3);

D13261 #40997283