#27162249

order	12625
automorphism	?
homology	Abelianization (H₁) = C₅² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₂₅×C₁₀₁ ⋊ C₅
Sylow	C₁₀₁ ⋊ 5₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini	C₅	C₅×C₁₀₁⋊C₅
commutator, socle, cosocle	C₅×C₁₀₁	C₅²
Fitting	C₂₅×C₁₀₁	C₅
prime radical	C₅×C₁₀₁⋊C₅	C₅
hypercenter	C₂₅	C₁₀₁⋊C₅
hypocenter	C₁₀₁	5₋¹⁺²
LCS	G ▷ C₅×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₅ ◁ C₂₅

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 2525

A minimal faithful permutation representation — 2 generators on 126 points:

(2,3,5,9,17)(4,7,13,25,14)(6,11,21,38,32)(8,15,28,50,76)(10,19,35,18,33)(12,23,42,49,27)(16,30,53,80,82)(20,37,61,46,29)(22,40,65,62,56)(24,44,54,48,74)(26,47,43,34,57)(31,39,63,59,36)(41,67,90,75,52)(45,71,93,91,98)(51,77,94,58,83)(55,66,88,87,86)(60,72,68,79,69)(64,78,85,73,70)(81,96,101,84,99)(89,97,100,95,92)(103,106,113,114,107)(105,110,111,112,120)(109,117,121,118,119)(116,123,124,125,126);(1,2,4,8,16,31,55,15,29,52,79,95,96,40,66,89,63,86,77,11,22,41,33,56,47,73,71,93,101,99,19,36,60,53,81,97,3,6,12,24,45,72,94,30,54,82,98,90,100,67,28,51,78,61,85,23,43,69,57,76,74,42,68,91,65,87,37,13,26,48,35,59,7,14,27,49,75,80,21,39,64,38,62,50,5,10,20,17,32,25,46,44,70,92,83,9,18,34,58,84,88)(102,103,105,109,116,104,107,111,118,123,108,114,120,117,124,115,113,110,119,125,122,106,112,121,126)

Polycyclic presentation:

5^3,101|a^5=c,b^a=bc,d^b=d^36

Conjugacy classes & character table

529 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12625, 11);