D₂₇₇₈₄ #72291111

order	55568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₇₈₄

Splittings

Fitting	C₈×C₂₃×C₁₅₁ ⋊ C₂
Hall	C₂₃×C₁₅₁ ⋊ D₈
Sylow	C₁₅₁ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₁₂₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₈₉₂
commutator, cosocle	C₄×C₂₃×C₁₅₁	C₂²
Frattini	C₄	C₂×D₃₄₇₃
Fitting	C₈×C₂₃×C₁₅₁	C₂
socle	C₂×C₂₃×C₁₅₁	D₄
hypercenter	C₈	D₃₄₇₃
hypocenter	C₂₃×C₁₅₁	D₈
LCS	G ▷ C₄×C₂₃×C₁₅₁ ▷ C₂×C₂₃×C₁₅₁ ▷ C₂₃×C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 182
transitive degree	μ_t(G) = 27784

A minimal faithful permutation representation — 2 generators on 182 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);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,182)(34,181)(35,180)(36,179)(37,178)(38,177)(39,176)(40,175)(41,174)(42,173)(43,172)(44,171)(45,170)(46,169)(47,168)(48,167)(49,166)(50,165)(51,164)(52,163)(53,162)(54,161)(55,160)(56,159)(57,158)(58,157)(59,156)(60,155)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)

Polycyclic presentation:

2^4,23,151|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^150

Conjugacy classes & character table

13895 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, 2, 2, 2, 23, 151]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^150);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27784 #72291111