D₁₅₀₂₈ #72236033

order	30056
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₀₂₈

Splittings

Fitting	C₄×C₁₃×C₂₈₉ ⋊ C₂
Hall	C₁₃×C₂₈₉ ⋊ D₄
Sylow	C₂₈₉ ⋊ D₅₂
Sylow	C₁₃ ⋊ D₁₁₅₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₇₅₇
commutator, cosocle	C₂×C₁₃×C₂₈₉	C₂²
Frattini	C₂×C₁₇	C₂×D₂₂₁
Fitting	C₄×C₁₃×C₂₈₉	C₂
socle	C₂×C₁₃×C₁₇	C₂×D₁₇
hypercenter	C₄	D₃₇₅₇
hypocenter	C₁₃×C₂₈₉	D₄
LCS	G ▷ C₂×C₁₃×C₂₈₉ ▷ C₁₃×C₂₈₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 306
transitive degree	μ_t(G) = 15028

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

Polycyclic presentation:

2^3,13,17^2|b^2=c,e^17=f,b^a=bc,d^a=d^12,e^a=e^16f^16,f^a=f^16

Conjugacy classes & character table

7517 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₀₅₆
maximal normal subgroup of	D₃₀₀₅₆
maximal quotient of	Dic₁₅₀₂₈, D₃₀₀₅₆

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, 13, 17, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^16*F.6^16);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15028 #72236033