Dic₈₉₄₀ #72273232

order	35760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₉₄₀

Splittings

Hall	C₃×C₅×C₁₄₉ ⋊ Q₁₆
Hall	C₅×C₁₄₉ ⋊ Dic₁₂
Hall	C₃×C₁₄₉ ⋊ Dic₂₀
Sylow	C₁₄₉ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₅₉₆
Sylow	C₅ ⋊ Dic₁₇₈₈
Sylow	C₃ ⋊ Dic₂₉₈₀

Subgroups

	subgroup	G/H
center	C₂	D₈₉₄₀
commutator, cosocle	C₃×C₄×C₅×C₁₄₉	C₂²
Frattini	C₄	C₂×D₂₂₃₅
Fitting	C₃×C₅×C₈×C₁₄₉	C₂
socle, prime radical	C₂×C₃×C₅×C₁₄₉	D₄
hypercenter	C₈	D₂₂₃₅
hypocenter	C₃×C₅×C₁₄₉	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₁₄₉ ▷ C₂×C₃×C₅×C₁₄₉ ▷ C₃×C₅×C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 173
transitive degree	μ_t(G) = 35760

A minimal faithful permutation representation — 2 generators on 173 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,173)(27,172)(28,171)(29,170)(30,169)(31,168)(32,167)(33,166)(34,165)(35,164)(36,163)(37,162)(38,161)(39,160)(40,159)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2^4,3,5,149|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^148

Conjugacy classes & character table

8943 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, 3, 5, 149]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^148);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8940 #72273232