Dic₁₆₃₂₈ #72301575

order	65312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₂₈

Splittings

Hall	C₁₃×C₁₅₇ ⋊ Q₃₂
Sylow	C₁₅₇ ⋊ Dic₁₀₄
Sylow	C₁₃ ⋊ Dic₁₂₅₆

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₂₈
commutator, cosocle	C₈×C₁₃×C₁₅₇	C₂²
Frattini	C₈	C₂×D₂₀₄₁
Fitting	C₁₃×C₁₆×C₁₅₇	C₂
socle, prime radical	C₂×C₁₃×C₁₅₇	D₈
hypercenter	C₁₆	D₂₀₄₁
hypocenter	C₁₃×C₁₅₇	Q₃₂
LCS	G ▷ C₈×C₁₃×C₁₅₇ ▷ C₄×C₁₃×C₁₅₇ ▷ C₂×C₁₃×C₁₅₇ ▷ C₁₃×C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 202
transitive degree	μ_t(G) = 65312

A minimal faithful permutation representation — 2 generators on 202 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);(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)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,202)(48,201)(49,200)(50,199)(51,198)(52,197)(53,196)(54,195)(55,194)(56,193)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,184)(66,183)(67,182)(68,181)(69,180)(70,179)(71,178)(72,177)(73,176)(74,175)(75,174)(76,173)(77,172)(78,171)(79,170)(80,169)(81,168)(82,167)(83,166)(84,165)(85,164)(86,163)(87,162)(88,161)(89,160)(90,159)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,150)(100,149)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)

Polycyclic presentation:

2^5,13,157|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,g^a=g^156

Conjugacy classes & character table

16331 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 13, 157]);;
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> 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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^156);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16328 #72301575