D₁₁₄₅₂ #72223903

order	22904
automorphism	Aut(G) order 56068992
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₄₅₂

Splittings

Fitting	C₄×C₇×C₄₀₉ ⋊ C₂
Hall	C₇×C₄₀₉ ⋊ D₄
Sylow	C₄₀₉ ⋊ D₂₈
Sylow	C₇ ⋊ D₁₆₃₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₈₆₃
commutator, socle, cosocle	C₂×C₇×C₄₀₉	C₂²
Fitting	C₄×C₇×C₄₀₉	C₂
hypercenter	C₄	D₂₈₆₃
hypocenter	C₇×C₄₀₉	D₄
LCS	G ▷ C₂×C₇×C₄₀₉ ▷ C₇×C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 420
transitive degree	μ_t(G) = 11452

A minimal faithful permutation representation — 2 generators on 420 points:

(1,2,3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420);(2,4)(6,11)(7,10)(8,9)(13,420)(14,419)(15,418)(16,417)(17,416)(18,415)(19,414)(20,413)(21,412)(22,411)(23,410)(24,409)(25,408)(26,407)(27,406)(28,405)(29,404)(30,403)(31,402)(32,401)(33,400)(34,399)(35,398)(36,397)(37,396)(38,395)(39,394)(40,393)(41,392)(42,391)(43,390)(44,389)(45,388)(46,387)(47,386)(48,385)(49,384)(50,383)(51,382)(52,381)(53,380)(54,379)(55,378)(56,377)(57,376)(58,375)(59,374)(60,373)(61,372)(62,371)(63,370)(64,369)(65,368)(66,367)(67,366)(68,365)(69,364)(70,363)(71,362)(72,361)(73,360)(74,359)(75,358)(76,357)(77,356)(78,355)(79,354)(80,353)(81,352)(82,351)(83,350)(84,349)(85,348)(86,347)(87,346)(88,345)(89,344)(90,343)(91,342)(92,341)(93,340)(94,339)(95,338)(96,337)(97,336)(98,335)(99,334)(100,333)(101,332)(102,331)(103,330)(104,329)(105,328)(106,327)(107,326)(108,325)(109,324)(110,323)(111,322)(112,321)(113,320)(114,319)(115,318)(116,317)(117,316)(118,315)(119,314)(120,313)(121,312)(122,311)(123,310)(124,309)(125,308)(126,307)(127,306)(128,305)(129,304)(130,303)(131,302)(132,301)(133,300)(134,299)(135,298)(136,297)(137,296)(138,295)(139,294)(140,293)(141,292)(142,291)(143,290)(144,289)(145,288)(146,287)(147,286)(148,285)(149,284)(150,283)(151,282)(152,281)(153,280)(154,279)(155,278)(156,277)(157,276)(158,275)(159,274)(160,273)(161,272)(162,271)(163,270)(164,269)(165,268)(166,267)(167,266)(168,265)(169,264)(170,263)(171,262)(172,261)(173,260)(174,259)(175,258)(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)

Polycyclic presentation:

2^3,7,409|b^2=c,b^a=bc,d^a=d^6,e^a=e^408

Conjugacy classes & character table

5729 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₉₀₄
maximal normal subgroup of	D₂₂₉₀₄
maximal quotient of	Dic₁₁₄₅₂, D₂₂₉₀₄

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, 7, 409]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^408);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11452 #72223903