#27162248

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

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, 10);