D₁₃₅₆₁ #41029396

order	27122
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₅₆₁
unambiguous semidirect product	C₇₁⋊D₁₉₁
unambiguous semidirect product	C₁₉₁⋊D₇₁

Splittings

Fitting, socle, derived, Hall	C₇₁×C₁₉₁ ⋊ C₂
Sylow	C₁₉₁ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₁₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₁×C₁₉₁	C₂
LCS	G ▷ C₇₁×C₁₉₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 262
transitive degree	μ_t(G) = 13561

A minimal faithful permutation representation — 2 generators on 262 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)(72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262);(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(73,262)(74,261)(75,260)(76,259)(77,258)(78,257)(79,256)(80,255)(81,254)(82,253)(83,252)(84,251)(85,250)(86,249)(87,248)(88,247)(89,246)(90,245)(91,244)(92,243)(93,242)(94,241)(95,240)(96,239)(97,238)(98,237)(99,236)(100,235)(101,234)(102,233)(103,232)(104,231)(105,230)(106,229)(107,228)(108,227)(109,226)(110,225)(111,224)(112,223)(113,222)(114,221)(115,220)(116,219)(117,218)(118,217)(119,216)(120,215)(121,214)(122,213)(123,212)(124,211)(125,210)(126,209)(127,208)(128,207)(129,206)(130,205)(131,204)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)

Polycyclic presentation:

2,71,191|b^a=b^70,c^a=c^190

Conjugacy classes & character table

6782 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₅₆₁
maximal normal subgroup of	C₂×D₁₃₅₆₁
maximal quotient of	Dic₁₃₅₆₁, C₂×D₁₃₅₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27122, 3);

D13561 #41029396