D₂₉₉₇₅ #72295323

order	59950
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₉₇₅

Splittings

Fitting, derived, Hall	C₁₁×C₂₅×C₁₀₉ ⋊ C₂
Hall	C₂₅×C₁₀₉ ⋊ D₁₁
Hall	C₁₁×C₁₀₉ ⋊ D₂₅
Hall	C₁₁×C₂₅ ⋊ D₁₀₉
Sylow	C₁₀₉ ⋊ D₂₇₅
Sylow	C₂₅ ⋊ D₁₁₉₉
Sylow	C₁₁ ⋊ D₂₇₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₁×C₂₅×C₁₀₉	C₂
Frattini	C₅	D₅₉₉₅
socle	C₅×C₁₁×C₁₀₉	D₅
LCS	G ▷ C₁₁×C₂₅×C₁₀₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 29975

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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,36)(28,35)(29,34)(30,33)(31,32)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2,5,11,109,5|b^5=e,b^a=b^4e^4,c^a=c^10,d^a=d^108,e^a=e^4

Conjugacy classes & character table

14989 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, 5, 11, 109, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^108);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29975 #72295323