Dic₁₃₅₆₀ #72289669

order	54240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₆₀

Splittings

Hall	C₃×C₅×C₁₁₃ ⋊ Q₃₂
Hall	C₅×C₁₁₃ ⋊ Dic₂₄
Hall	C₃×C₁₁₃ ⋊ Dic₄₀
Sylow	C₁₁₃ ⋊ Dic₁₂₀
Hall	C₃×C₅ ⋊ Dic₉₀₄
Sylow	C₅ ⋊ Dic₂₇₁₂
Sylow	C₃ ⋊ Dic₄₅₂₀

Subgroups

	subgroup	G/H
center	C₂	D₁₃₅₆₀
commutator, cosocle	C₃×C₅×C₈×C₁₁₃	C₂²
Frattini	C₈	C₂×D₁₆₉₅
Fitting	C₃×C₅×C₁₆×C₁₁₃	C₂
socle, prime radical	C₂×C₃×C₅×C₁₁₃	D₈
hypercenter	C₁₆	D₁₆₉₅
hypocenter	C₃×C₅×C₁₁₃	Q₃₂
LCS	G ▷ C₃×C₅×C₈×C₁₁₃ ▷ C₃×C₄×C₅×C₁₁₃ ▷ C₂×C₃×C₅×C₁₁₃ ▷ C₃×C₅×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 153
transitive degree	μ_t(G) = 54240

A minimal faithful permutation representation — 2 generators on 153 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);(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)(34,35)(37,40)(38,39)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^5,3,5,113|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^4,h^a=h^112

Conjugacy classes & character table

13563 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, 5, 113]);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13560 #72289669