D₂₇₀₆₈ #72290399

order	54136
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₀₆₈

Splittings

Fitting	C₄×C₆₇×C₁₀₁ ⋊ C₂
Hall	C₆₇×C₁₀₁ ⋊ D₄
Sylow	C₁₀₁ ⋊ D₂₆₈
Sylow	C₆₇ ⋊ D₄₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₇₆₇
commutator, socle, cosocle	C₂×C₆₇×C₁₀₁	C₂²
Fitting	C₄×C₆₇×C₁₀₁	C₂
hypercenter	C₄	D₆₇₆₇
hypocenter	C₆₇×C₁₀₁	D₄
LCS	G ▷ C₂×C₆₇×C₁₀₁ ▷ C₆₇×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 27068

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

Polycyclic presentation:

2^3,67,101|b^2=c,b^a=bc,d^a=d^66,e^a=e^100

Conjugacy classes & character table

13537 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, 67, 101]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^66);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27068 #72290399