D₃₀₁₅₆ #72295464

order	60312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₅₆

Splittings

Fitting	C₃×C₄×C₇×C₃₅₉ ⋊ C₂
Hall	C₃×C₇×C₃₅₉ ⋊ D₄
Hall	C₇×C₃₅₉ ⋊ D₁₂
Hall	C₃×C₃₅₉ ⋊ D₂₈
Sylow	C₃₅₉ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₁₄₃₆
Sylow	C₇ ⋊ D₄₃₀₈
Sylow	C₃ ⋊ D₁₀₀₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₃₉
commutator, socle, cosocle	C₂×C₃×C₇×C₃₅₉	C₂²
Fitting	C₃×C₄×C₇×C₃₅₉	C₂
hypercenter	C₄	D₇₅₃₉
hypocenter	C₃×C₇×C₃₅₉	D₄
LCS	G ▷ C₂×C₃×C₇×C₃₅₉ ▷ C₃×C₇×C₃₅₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 373
transitive degree	μ_t(G) = 30156

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

Polycyclic presentation:

2^3,3,7,359|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^358

Conjugacy classes & character table

15081 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 359]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^358);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30156 #72295464