D₂₇₉₂₇ #72291245

order	55854
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₇₉₂₇

Splittings

Fitting, derived, Hall	C₉×C₂₉×C₁₀₇ ⋊ C₂
Hall	C₂₉×C₁₀₇ ⋊ D₉
Hall	C₉×C₁₀₇ ⋊ D₂₉
Hall	C₉×C₂₉ ⋊ D₁₀₇
Sylow	C₁₀₇ ⋊ D₂₆₁
Sylow	C₂₉ ⋊ D₉₆₃
Sylow	C₉ ⋊ D₃₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₉×C₁₀₇	C₂
Frattini	C₃	D₉₃₀₉
socle	C₃×C₂₉×C₁₀₇	D₃
LCS	G ▷ C₉×C₂₉×C₁₀₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 27927

A minimal faithful permutation representation — 2 generators on 145 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);(2,9)(3,8)(4,7)(5,6)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)

Polycyclic presentation:

2,3,29,107,3|b^3=e,b^a=b^2e^2,c^a=c^28,d^a=d^106,e^a=e^2

Conjugacy classes & character table

13965 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, 3, 29, 107, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^28);;
gap> SetConjugate(coll, 4, 1, F.4^106);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27927 #72291245