Dic₈₁₁₂ #72272500

order	32448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₁₁₂

Splittings

Hall	C₃×C₁₆₉ ⋊ Q₆₄
Sylow	C₁₆₉ ⋊ Dic₄₈
Sylow	C₃ ⋊ Dic₂₇₀₄

Subgroups

	subgroup	G/H
center	C₂	D₈₁₁₂
commutator, cosocle	C₃×C₁₆×C₁₆₉	C₂²
Frattini	C₁₃×C₁₆	C₂×D₃₉
Fitting	C₃×C₃₂×C₁₆₉	C₂
socle, prime radical	C₂×C₃×C₁₃	D₂₀₈
hypercenter	C₃₂	D₅₀₇
hypocenter	C₃×C₁₆₉	Q₆₄
LCS	G ▷ C₃×C₁₆×C₁₆₉ ▷ C₃×C₈×C₁₆₉ ▷ C₃×C₄×C₁₆₉ ▷ C₂×C₃×C₁₆₉ ▷ C₃×C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 32448

A minimal faithful permutation representation — 2 generators on 236 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,67)(69,236)(70,235)(71,234)(72,233)(73,232)(74,231)(75,230)(76,229)(77,228)(78,227)(79,226)(80,225)(81,224)(82,223)(83,222)(84,221)(85,220)(86,219)(87,218)(88,217)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)(117,188)(118,187)(119,186)(120,185)(121,184)(122,183)(123,182)(124,181)(125,180)(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)(132,173)(133,172)(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)(140,165)(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)

Polycyclic presentation:

2^6,3,13^2|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,h^13=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^12i^12,i^a=i^12

Conjugacy classes & character table

8115 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₂₂₄
maximal normal subgroup of	Dic₁₆₂₂₄

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, 2, 2, 2, 3, 13, 13]);;
gap> SetPower(coll, 1, F.6);;
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);;
gap> SetPower(coll, 8, F.9);;
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, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^12*F.9^12);;
gap> SetConjugate(coll, 9, 1, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8112 #72272500