D₂₈₆₁₂ #72292233

order	57224
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₆₁₂

Splittings

Fitting	C₄×C₂₃×C₃₁₁ ⋊ C₂
Hall	C₂₃×C₃₁₁ ⋊ D₄
Sylow	C₃₁₁ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₁₂₄₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₁₅₃
commutator, socle, cosocle	C₂×C₂₃×C₃₁₁	C₂²
Fitting	C₄×C₂₃×C₃₁₁	C₂
hypercenter	C₄	D₇₁₅₃
hypocenter	C₂₃×C₃₁₁	D₄
LCS	G ▷ C₂×C₂₃×C₃₁₁ ▷ C₂₃×C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 338
transitive degree	μ_t(G) = 28612

A minimal faithful permutation representation — 2 generators on 338 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338);(2,4)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(29,338)(30,337)(31,336)(32,335)(33,334)(34,333)(35,332)(36,331)(37,330)(38,329)(39,328)(40,327)(41,326)(42,325)(43,324)(44,323)(45,322)(46,321)(47,320)(48,319)(49,318)(50,317)(51,316)(52,315)(53,314)(54,313)(55,312)(56,311)(57,310)(58,309)(59,308)(60,307)(61,306)(62,305)(63,304)(64,303)(65,302)(66,301)(67,300)(68,299)(69,298)(70,297)(71,296)(72,295)(73,294)(74,293)(75,292)(76,291)(77,290)(78,289)(79,288)(80,287)(81,286)(82,285)(83,284)(84,283)(85,282)(86,281)(87,280)(88,279)(89,278)(90,277)(91,276)(92,275)(93,274)(94,273)(95,272)(96,271)(97,270)(98,269)(99,268)(100,267)(101,266)(102,265)(103,264)(104,263)(105,262)(106,261)(107,260)(108,259)(109,258)(110,257)(111,256)(112,255)(113,254)(114,253)(115,252)(116,251)(117,250)(118,249)(119,248)(120,247)(121,246)(122,245)(123,244)(124,243)(125,242)(126,241)(127,240)(128,239)(129,238)(130,237)(131,236)(132,235)(133,234)(134,233)(135,232)(136,231)(137,230)(138,229)(139,228)(140,227)(141,226)(142,225)(143,224)(144,223)(145,222)(146,221)(147,220)(148,219)(149,218)(150,217)(151,216)(152,215)(153,214)(154,213)(155,212)(156,211)(157,210)(158,209)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)

Polycyclic presentation:

2^3,23,311|b^2=c,b^a=bc,d^a=d^22,e^a=e^310

Conjugacy classes & character table

14309 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 23, 311]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^310);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28612 #72292233