D₃₀₁₂ #72210776

order	6024
automorphism	Aut(G) order 3012000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₂

Splittings

Fitting	C₃×C₄×C₂₅₁ ⋊ C₂
Hall	C₃×C₂₅₁ ⋊ D₄
Sylow	C₂₅₁ ⋊ D₁₂
Sylow	C₃ ⋊ D₁₀₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₃
commutator, socle, cosocle	C₂×C₃×C₂₅₁	C₂²
Fitting	C₃×C₄×C₂₅₁	C₂
hypercenter	C₄	D₇₅₃
hypocenter	C₃×C₂₅₁	D₄
LCS	G ▷ C₂×C₃×C₂₅₁ ▷ C₃×C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 258
transitive degree	μ_t(G) = 3012

A minimal faithful permutation representation — 2 generators on 258 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);(2,4)(6,7)(9,258)(10,257)(11,256)(12,255)(13,254)(14,253)(15,252)(16,251)(17,250)(18,249)(19,248)(20,247)(21,246)(22,245)(23,244)(24,243)(25,242)(26,241)(27,240)(28,239)(29,238)(30,237)(31,236)(32,235)(33,234)(34,233)(35,232)(36,231)(37,230)(38,229)(39,228)(40,227)(41,226)(42,225)(43,224)(44,223)(45,222)(46,221)(47,220)(48,219)(49,218)(50,217)(51,216)(52,215)(53,214)(54,213)(55,212)(56,211)(57,210)(58,209)(59,208)(60,207)(61,206)(62,205)(63,204)(64,203)(65,202)(66,201)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,186)(82,185)(83,184)(84,183)(85,182)(86,181)(87,180)(88,179)(89,178)(90,177)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)

Polycyclic presentation:

2^3,3,251|b^2=c,b^a=bc,d^a=d^2,e^a=e^250

Conjugacy classes & character table

1509 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₀₂₄, D₉₀₃₆, D₁₅₀₆₀, …
maximal normal subgroup of	D₆₀₂₄
maximal quotient of	D₆₀₂₄, Dic₃₀₁₂, D₉₀₃₆, …

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, 3, 251]);;
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.5^250);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3012 #72210776