D₂₆₂₇₁ #72289632

order	52542
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₂₇₁

Splittings

Fitting, derived, Hall	C₇×C₂₇×C₁₃₉ ⋊ C₂
Hall	C₂₇×C₁₃₉ ⋊ D₇
Hall	C₇×C₁₃₉ ⋊ D₂₇
Hall	C₇×C₂₇ ⋊ D₁₃₉
Sylow	C₁₃₉ ⋊ D₁₈₉
Sylow	C₂₇ ⋊ D₉₇₃
Sylow	C₇ ⋊ D₃₇₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₇×C₁₃₉	C₂
Frattini	C₉	D₂₉₁₉
socle	C₃×C₇×C₁₃₉	D₉
LCS	G ▷ C₇×C₂₇×C₁₃₉
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 173
transitive degree	μ_t(G) = 26271

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

Polycyclic presentation:

2,3,7,139,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^6,d^a=d^138,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

13137 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", "f");;
gap> coll := SingleCollector(F, [2, 3, 7, 139, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^138);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26271 #72289632