D₂₇₃₈₈ #72290696

order	54776
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₃₈₈

Splittings

Fitting	C₄×C₄₁×C₁₆₇ ⋊ C₂
Hall	C₄₁×C₁₆₇ ⋊ D₄
Sylow	C₁₆₇ ⋊ D₁₆₄
Sylow	C₄₁ ⋊ D₆₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₄₇
commutator, socle, cosocle	C₂×C₄₁×C₁₆₇	C₂²
Fitting	C₄×C₄₁×C₁₆₇	C₂
hypercenter	C₄	D₆₈₄₇
hypocenter	C₄₁×C₁₆₇	D₄
LCS	G ▷ C₂×C₄₁×C₁₆₇ ▷ C₄₁×C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 212
transitive degree	μ_t(G) = 27388

A minimal faithful permutation representation — 2 generators on 212 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212);(2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(47,212)(48,211)(49,210)(50,209)(51,208)(52,207)(53,206)(54,205)(55,204)(56,203)(57,202)(58,201)(59,200)(60,199)(61,198)(62,197)(63,196)(64,195)(65,194)(66,193)(67,192)(68,191)(69,190)(70,189)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(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:

2^3,41,167|b^2=c,b^a=bc,d^a=d^40,e^a=e^166

Conjugacy classes & character table

13697 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, 41, 167]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27388 #72290696