D₃₀₁₈₄ #72295488

order	60368
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₈₄

Splittings

Fitting	C₈×C₁₁×C₃₄₃ ⋊ C₂
Hall	C₁₁×C₃₄₃ ⋊ D₈
Sylow	C₃₄₃ ⋊ D₈₈
Sylow	C₁₁ ⋊ D₂₇₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₀₉₂
commutator, cosocle	C₄×C₁₁×C₃₄₃	C₂²
Frattini	C₄×C₄₉	C₂×D₇₇
Fitting	C₈×C₁₁×C₃₄₃	C₂
socle	C₂×C₇×C₁₁	D₁₉₆
hypercenter	C₈	D₃₇₇₃
hypocenter	C₁₁×C₃₄₃	D₈
LCS	G ▷ C₄×C₁₁×C₃₄₃ ▷ C₂×C₁₁×C₃₄₃ ▷ C₁₁×C₃₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 362
transitive degree	μ_t(G) = 30184

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

Polycyclic presentation:

2^4,7,11,7^2|b^2=cd,c^2=d,e^7=g,g^7=h,b^a=bc,c^a=cd,e^a=e^6g^6h^6,f^a=f^10,g^a=g^6h^6,h^a=h^6

Conjugacy classes & character table

15095 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 7, 11, 7, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6*F.8^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^6*F.8^6);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30184 #72295488