D₂₆₂₂₈ #72289588

order	52456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₂₂₈

Splittings

Fitting	C₄×C₇₉×C₈₃ ⋊ C₂
Hall	C₇₉×C₈₃ ⋊ D₄
Sylow	C₈₃ ⋊ D₃₁₆
Sylow	C₇₉ ⋊ D₃₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₅₅₇
commutator, socle, cosocle	C₂×C₇₉×C₈₃	C₂²
Fitting	C₄×C₇₉×C₈₃	C₂
hypercenter	C₄	D₆₅₅₇
hypocenter	C₇₉×C₈₃	D₄
LCS	G ▷ C₂×C₇₉×C₈₃ ▷ C₇₉×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 166
transitive degree	μ_t(G) = 26228

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

Polycyclic presentation:

2^3,79,83|b^2=c,b^a=bc,d^a=d^78,e^a=e^82

Conjugacy classes & character table

13117 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, 2, 79, 83]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^78);;
gap> SetConjugate(coll, 5, 1, F.5^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26228 #72289588