D₁₄₂₅₇ #43250026

order	28514
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₄₂₅₇
unambiguous semidirect product	C₅₃⋊D₂₆₉
unambiguous semidirect product	C₂₆₉⋊D₅₃

Splittings

Fitting, socle, derived, Hall	C₅₃×C₂₆₉ ⋊ C₂
Sylow	C₂₆₉ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅₃×C₂₆₉	C₂
LCS	G ▷ C₅₃×C₂₆₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 322
transitive degree	μ_t(G) = 14257

A minimal faithful permutation representation — 2 generators on 322 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)(54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322);(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(55,322)(56,321)(57,320)(58,319)(59,318)(60,317)(61,316)(62,315)(63,314)(64,313)(65,312)(66,311)(67,310)(68,309)(69,308)(70,307)(71,306)(72,305)(73,304)(74,303)(75,302)(76,301)(77,300)(78,299)(79,298)(80,297)(81,296)(82,295)(83,294)(84,293)(85,292)(86,291)(87,290)(88,289)(89,288)(90,287)(91,286)(92,285)(93,284)(94,283)(95,282)(96,281)(97,280)(98,279)(99,278)(100,277)(101,276)(102,275)(103,274)(104,273)(105,272)(106,271)(107,270)(108,269)(109,268)(110,267)(111,266)(112,265)(113,264)(114,263)(115,262)(116,261)(117,260)(118,259)(119,258)(120,257)(121,256)(122,255)(123,254)(124,253)(125,252)(126,251)(127,250)(128,249)(129,248)(130,247)(131,246)(132,245)(133,244)(134,243)(135,242)(136,241)(137,240)(138,239)(139,238)(140,237)(141,236)(142,235)(143,234)(144,233)(145,232)(146,231)(147,230)(148,229)(149,228)(150,227)(151,226)(152,225)(153,224)(154,223)(155,222)(156,221)(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)(164,213)(165,212)(166,211)(167,210)(168,209)(169,208)(170,207)(171,206)(172,205)(173,204)(174,203)(175,202)(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)(182,195)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)

Polycyclic presentation:

2,53,269|b^a=b^52,c^a=c^268

Conjugacy classes & character table

7130 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₄₂₅₇
maximal normal subgroup of	C₂×D₁₄₂₅₇
maximal quotient of	Dic₁₄₂₅₇, C₂×D₁₄₂₅₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28514, 3);

D14257 #43250026