Dic₁₃₁₁₄ #72288714

order	52456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₁₁₄

Splittings

Hall	C₇₉×C₈₃ ⋊ Q₈
Sylow	C₈₃ ⋊ Dic₁₅₈
Sylow	C₇₉ ⋊ Dic₁₆₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₅₅₇
commutator, socle, prime radical, cosocle	C₂×C₇₉×C₈₃	C₂²
Fitting	C₄×C₇₉×C₈₃	C₂
hypercenter	C₄	D₆₅₅₇
hypocenter	C₇₉×C₈₃	Q₈
LCS	G ▷ C₂×C₇₉×C₈₃ ▷ C₇₉×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 170
transitive degree	μ_t(G) = 52456

A minimal faithful permutation representation — 2 generators on 170 points:

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

Polycyclic presentation:

2^3,79,83|a^2=c,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, 1, F.3);;
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);

Dic13114 #72288714