D₆₈₀₄ #72212401

order	13608
automorphism	Aut(G) order 13226976
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₈₀₄

Splittings

Fitting	C₄×C₇×C₂₄₃ ⋊ C₂
Hall	C₇×C₂₄₃ ⋊ D₄
Sylow	C₂₄₃ ⋊ D₂₈
Sylow	C₇ ⋊ D₉₇₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₇₀₁
commutator, cosocle	C₂×C₇×C₂₄₃	C₂²
Frattini	C₂×C₈₁	C₂×D₂₁
Fitting	C₄×C₇×C₂₄₃	C₂
socle	C₂×C₃×C₇	C₂×D₈₁
hypercenter	C₄	D₁₇₀₁
hypocenter	C₇×C₂₄₃	D₄
LCS	G ▷ C₂×C₇×C₂₄₃ ▷ C₇×C₂₄₃
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 254
transitive degree	μ_t(G) = 6804

A minimal faithful permutation representation — 2 generators on 254 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);(2,4)(6,247)(7,246)(8,245)(9,244)(10,243)(11,242)(12,241)(13,240)(14,239)(15,238)(16,237)(17,236)(18,235)(19,234)(20,233)(21,232)(22,231)(23,230)(24,229)(25,228)(26,227)(27,226)(28,225)(29,224)(30,223)(31,222)(32,221)(33,220)(34,219)(35,218)(36,217)(37,216)(38,215)(39,214)(40,213)(41,212)(42,211)(43,210)(44,209)(45,208)(46,207)(47,206)(48,205)(49,204)(50,203)(51,202)(52,201)(53,200)(54,199)(55,198)(56,197)(57,196)(58,195)(59,194)(60,193)(61,192)(62,191)(63,190)(64,189)(65,188)(66,187)(67,186)(68,185)(69,184)(70,183)(71,182)(72,181)(73,180)(74,179)(75,178)(76,177)(77,176)(78,175)(79,174)(80,173)(81,172)(82,171)(83,170)(84,169)(85,168)(86,167)(87,166)(88,165)(89,164)(90,163)(91,162)(92,161)(93,160)(94,159)(95,158)(96,157)(97,156)(98,155)(99,154)(100,153)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(249,254)(250,253)(251,252)

Polycyclic presentation:

2^3,3,7,3^4|b^2=c,d^3=f,f^3=g,g^3=h,h^3=i,b^a=bc,d^a=d^2f^2g^2h^2i^2,e^a=e^6,f^a=f^2g^2h^2i^2,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

3405 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₆₀₈, D₂₀₄₁₂
maximal normal subgroup of	D₁₃₆₀₈
maximal quotient of	D₁₃₆₀₈, Dic₆₈₀₄, D₂₀₄₁₂

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6804 #72212401