Dic₁₃₂₆₈ #72288965

order	53072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₂₆₈

Splittings

Hall	C₃₁×C₁₀₇ ⋊ Q₁₆
Sylow	C₁₀₇ ⋊ Dic₁₂₄
Sylow	C₃₁ ⋊ Dic₄₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₂₆₈
commutator, cosocle	C₄×C₃₁×C₁₀₇	C₂²
Frattini	C₄	C₂×D₃₃₁₇
Fitting	C₈×C₃₁×C₁₀₇	C₂
socle, prime radical	C₂×C₃₁×C₁₀₇	D₄
hypercenter	C₈	D₃₃₁₇
hypocenter	C₃₁×C₁₀₇	Q₁₆
LCS	G ▷ C₄×C₃₁×C₁₀₇ ▷ C₂×C₃₁×C₁₀₇ ▷ C₃₁×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 154
transitive degree	μ_t(G) = 53072

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

Polycyclic presentation:

2^4,31,107|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^30,f^a=f^106

Conjugacy classes & character table

13271 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 31, 107]);;
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.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13268 #72288965