D₂₆₇₈₈ #72290131

order	53576
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₇₈₈

Splittings

Fitting	C₄×C₃₇×C₁₈₁ ⋊ C₂
Hall	C₃₇×C₁₈₁ ⋊ D₄
Sylow	C₁₈₁ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₇₂₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₉₇
commutator, socle, cosocle	C₂×C₃₇×C₁₈₁	C₂²
Fitting	C₄×C₃₇×C₁₈₁	C₂
hypercenter	C₄	D₆₆₉₇
hypocenter	C₃₇×C₁₈₁	D₄
LCS	G ▷ C₂×C₃₇×C₁₈₁ ▷ C₃₇×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 222
transitive degree	μ_t(G) = 26788

A minimal faithful permutation representation — 2 generators on 222 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222);(2,4)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(43,222)(44,221)(45,220)(46,219)(47,218)(48,217)(49,216)(50,215)(51,214)(52,213)(53,212)(54,211)(55,210)(56,209)(57,208)(58,207)(59,206)(60,205)(61,204)(62,203)(63,202)(64,201)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,192)(74,191)(75,190)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(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:

2^3,37,181|b^2=c,b^a=bc,d^a=d^36,e^a=e^180

Conjugacy classes & character table

13397 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, 37, 181]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26788 #72290131