Dic₆₇₆₀ #72236282

order	27040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₇₆₀

Splittings

Hall	C₅×C₁₆₉ ⋊ Q₃₂
Sylow	C₁₆₉ ⋊ Dic₄₀
Sylow	C₅ ⋊ Dic₁₃₅₂

Subgroups

	subgroup	G/H
center	C₂	D₆₇₆₀
commutator, cosocle	C₅×C₈×C₁₆₉	C₂²
Frattini	C₈×C₁₃	C₂×D₆₅
Fitting	C₅×C₁₆×C₁₆₉	C₂
socle, prime radical	C₂×C₅×C₁₃	D₁₀₄
hypercenter	C₁₆	D₈₄₅
hypocenter	C₅×C₁₆₉	Q₃₂
LCS	G ▷ C₅×C₈×C₁₆₉ ▷ C₄×C₅×C₁₆₉ ▷ C₂×C₅×C₁₆₉ ▷ C₅×C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 206
transitive degree	μ_t(G) = 27040

A minimal faithful permutation representation — 2 generators on 206 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);(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)(34,37)(35,36)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,192)(54,191)(55,190)(56,189)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,164)(82,163)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)

Polycyclic presentation:

2^5,5,13^2|a^2=e,b^2=cd,c^2=de,d^2=e,g^13=h,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^12h^12,h^a=h^12

Conjugacy classes & character table

6763 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₅₂₀
maximal normal subgroup of	Dic₁₃₅₂₀

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, 5, 13, 13]);;
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, 7, 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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^12*F.8^12);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6760 #72236282