Dic₁₂₉₂₇ #72288415

order	51708
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₉₂₇

Splittings

derived, Hall	C₃×C₃₁×C₁₃₉ ⋊ C₄
Hall	C₃₁×C₁₃₉ ⋊ Dic₃
Hall	C₃×C₁₃₉ ⋊ Dic₃₁
Sylow	C₁₃₉ ⋊ Dic₉₃
Hall	C₃×C₃₁ ⋊ Dic₁₃₉
Sylow	C₃₁ ⋊ Dic₄₁₇
Sylow	C₃ ⋊ Dic₄₃₀₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₉₂₇
commutator, hypocenter	C₃×C₃₁×C₁₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₁×C₁₃₉	C₂
LCS	G ▷ C₃×C₃₁×C₁₃₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 177
transitive degree	μ_t(G) = 51708

A minimal faithful permutation representation — 2 generators on 177 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(1,3,2,4)(6,7)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(40,177)(41,176)(42,175)(43,174)(44,173)(45,172)(46,171)(47,170)(48,169)(49,168)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,161)(57,160)(58,159)(59,158)(60,157)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,130)(88,129)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)

Polycyclic presentation:

2^2,3,31,139|a^2=b,c^a=c^2,d^a=d^30,e^a=e^138

Conjugacy classes & character table

12930 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 31, 139]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12927 #72288415