D₂₆₈₅₆ #72290192

order	53712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₈₅₆

Splittings

Fitting	C₈×C₉×C₃₇₃ ⋊ C₂
Hall	C₉×C₃₇₃ ⋊ D₈
Sylow	C₃₇₃ ⋊ D₇₂
Sylow	C₉ ⋊ D₂₉₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₄₂₈
commutator, cosocle	C₄×C₉×C₃₇₃	C₂²
Frattini	C₃×C₄	C₂×D₁₁₁₉
Fitting	C₈×C₉×C₃₇₃	C₂
socle	C₂×C₃×C₃₇₃	D₁₂
hypercenter	C₈	D₃₃₅₇
hypocenter	C₉×C₃₇₃	D₈
LCS	G ▷ C₄×C₉×C₃₇₃ ▷ C₂×C₉×C₃₇₃ ▷ C₉×C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 390
transitive degree	μ_t(G) = 26856

A minimal faithful permutation representation — 2 generators on 390 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,390)(20,389)(21,388)(22,387)(23,386)(24,385)(25,384)(26,383)(27,382)(28,381)(29,380)(30,379)(31,378)(32,377)(33,376)(34,375)(35,374)(36,373)(37,372)(38,371)(39,370)(40,369)(41,368)(42,367)(43,366)(44,365)(45,364)(46,363)(47,362)(48,361)(49,360)(50,359)(51,358)(52,357)(53,356)(54,355)(55,354)(56,353)(57,352)(58,351)(59,350)(60,349)(61,348)(62,347)(63,346)(64,345)(65,344)(66,343)(67,342)(68,341)(69,340)(70,339)(71,338)(72,337)(73,336)(74,335)(75,334)(76,333)(77,332)(78,331)(79,330)(80,329)(81,328)(82,327)(83,326)(84,325)(85,324)(86,323)(87,322)(88,321)(89,320)(90,319)(91,318)(92,317)(93,316)(94,315)(95,314)(96,313)(97,312)(98,311)(99,310)(100,309)(101,308)(102,307)(103,306)(104,305)(105,304)(106,303)(107,302)(108,301)(109,300)(110,299)(111,298)(112,297)(113,296)(114,295)(115,294)(116,293)(117,292)(118,291)(119,290)(120,289)(121,288)(122,287)(123,286)(124,285)(125,284)(126,283)(127,282)(128,281)(129,280)(130,279)(131,278)(132,277)(133,276)(134,275)(135,274)(136,273)(137,272)(138,271)(139,270)(140,269)(141,268)(142,267)(143,266)(144,265)(145,264)(146,263)(147,262)(148,261)(149,260)(150,259)(151,258)(152,257)(153,256)(154,255)(155,254)(156,253)(157,252)(158,251)(159,250)(160,249)(161,248)(162,247)(163,246)(164,245)(165,244)(166,243)(167,242)(168,241)(169,240)(170,239)(171,238)(172,237)(173,236)(174,235)(175,234)(176,233)(177,232)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)(184,225)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)(200,209)(201,208)(202,207)(203,206)(204,205)

Polycyclic presentation:

2^4,3,373,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^372,g^a=g^2

Conjugacy classes & character table

13431 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", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 373, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^372);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26856 #72290192