Dic₂₅₉₂ #72212846

order	10368
automorphism	Aut(G) order 8957952
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₅₉₂

Splittings

Sylow

C₈₁ ⋊ Q₁₂₈

Subgroups

	subgroup	G/H
center	C₂	D₂₅₉₂
commutator, cosocle	C₃₂×C₈₁	C₂²
Frattini	C₂₇×C₃₂	C₂×D₃
Fitting	C₆₄×C₈₁	C₂
socle, prime radical	C₂×C₃	D₈₆₄
hypercenter	C₆₄	D₈₁
hypocenter	C₈₁	Q₁₂₈
LCS	G ▷ C₃₂×C₈₁ ▷ C₁₆×C₈₁ ▷ C₈×C₈₁ ▷ C₄×C₈₁ ▷ C₂×C₈₁ ▷ C₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 10368

A minimal faithful permutation representation — 2 generators on 209 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(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);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,209)(131,208)(132,207)(133,206)(134,205)(135,204)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,189)(151,188)(152,187)(153,186)(154,185)(155,184)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)

Polycyclic presentation:

2^7,3^4|a^2=g,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^3=i,i^3=j,j^3=k,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2i^2j^2k^2,i^a=i^2j^2k^2,j^a=j^2k^2,k^a=k^2

Conjugacy classes & character table

2595 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₁₈₄, Dic₇₇₇₆, Dic₁₂₉₆₀
maximal normal subgroup of	Dic₅₁₈₄
maximal quotient of	Dic₇₇₇₆, Dic₁₂₉₆₀

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2592 #72212846