Dic₁₅₇₄₈ #72295600

order	62992
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₇₄₈

Splittings

Hall	C₃₁×C₁₂₇ ⋊ Q₁₆
Sylow	C₁₂₇ ⋊ Dic₁₂₄
Sylow	C₃₁ ⋊ Dic₅₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₄₈
commutator, cosocle	C₄×C₃₁×C₁₂₇	C₂²
Frattini	C₄	C₂×D₃₉₃₇
Fitting	C₈×C₃₁×C₁₂₇	C₂
socle, prime radical	C₂×C₃₁×C₁₂₇	D₄
hypercenter	C₈	D₃₉₃₇
hypocenter	C₃₁×C₁₂₇	Q₁₆
LCS	G ▷ C₄×C₃₁×C₁₂₇ ▷ C₂×C₃₁×C₁₂₇ ▷ C₃₁×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 174
transitive degree	μ_t(G) = 62992

A minimal faithful permutation representation — 2 generators on 174 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)(48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174);(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)(49,174)(50,173)(51,172)(52,171)(53,170)(54,169)(55,168)(56,167)(57,166)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)

Polycyclic presentation:

2^4,31,127|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^30,f^a=f^126

Conjugacy classes & character table

15751 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, 127]);;
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.6^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15748 #72295600