Dic₁₄₃₂₈ #72290937

order	57312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₂₈

Splittings

Hall	C₉×C₁₉₉ ⋊ Q₃₂
Sylow	C₁₉₉ ⋊ Dic₇₂
Sylow	C₉ ⋊ Dic₁₅₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₃₂₈
commutator, cosocle	C₈×C₉×C₁₉₉	C₂²
Frattini	C₃×C₈	C₂×D₅₉₇
Fitting	C₉×C₁₆×C₁₉₉	C₂
socle, prime radical	C₂×C₃×C₁₉₉	D₂₄
hypercenter	C₁₆	D₁₇₉₁
hypocenter	C₉×C₁₉₉	Q₃₂
LCS	G ▷ C₈×C₉×C₁₉₉ ▷ C₄×C₉×C₁₉₉ ▷ C₂×C₉×C₁₉₉ ▷ C₉×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 240
transitive degree	μ_t(G) = 57312

A minimal faithful permutation representation — 2 generators on 240 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,41)(35,40)(36,39)(37,38)(43,240)(44,239)(45,238)(46,237)(47,236)(48,235)(49,234)(50,233)(51,232)(52,231)(53,230)(54,229)(55,228)(56,227)(57,226)(58,225)(59,224)(60,223)(61,222)(62,221)(63,220)(64,219)(65,218)(66,217)(67,216)(68,215)(69,214)(70,213)(71,212)(72,211)(73,210)(74,209)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,182)(102,181)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175)(109,174)(110,173)(111,172)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,165)(119,164)(120,163)(121,162)(122,161)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)

Polycyclic presentation:

2^5,3,199,3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^198,h^a=h^2

Conjugacy classes & character table

14331 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 199, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
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, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^198);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14328 #72290937