D₃₀₆₅₂ #72295965

order	61304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₆₅₂

Splittings

Fitting	C₄×C₇₉×C₉₇ ⋊ C₂
Hall	C₇₉×C₉₇ ⋊ D₄
Sylow	C₉₇ ⋊ D₃₁₆
Sylow	C₇₉ ⋊ D₃₈₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₆₆₃
commutator, socle, cosocle	C₂×C₇₉×C₉₇	C₂²
Fitting	C₄×C₇₉×C₉₇	C₂
hypercenter	C₄	D₇₆₆₃
hypocenter	C₇₉×C₉₇	D₄
LCS	G ▷ C₂×C₇₉×C₉₇ ▷ C₇₉×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 180
transitive degree	μ_t(G) = 30652

A minimal faithful permutation representation — 2 generators on 180 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)(84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180);(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)(85,180)(86,179)(87,178)(88,177)(89,176)(90,175)(91,174)(92,173)(93,172)(94,171)(95,170)(96,169)(97,168)(98,167)(99,166)(100,165)(101,164)(102,163)(103,162)(104,161)(105,160)(106,159)(107,158)(108,157)(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,147)(119,146)(120,145)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)

Polycyclic presentation:

2^3,79,97|b^2=c,b^a=bc,d^a=d^78,e^a=e^96

Conjugacy classes & character table

15329 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, 97]);;
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.5^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30652 #72295965