#27187702

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	#131×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #2875

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂³×#87938
commutator, socle	C₂×C₁₀₁	C₂²×C₄²
Fitting	#131×C₁₀₁	C₄
prime radical	#41×D₁₀₁	C₂²
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	#131	#87938
hypocenter	C₁₀₁	#2875
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ #131

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 1616

A minimal faithful permutation representation — 4 generators on 117 points:

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

Polycyclic presentation:

2^7,101|a^2=e,b^2=f,e^2=g,b^a=bg,h^b=h^10,d^c=dg,h^f=h^100

Conjugacy classes & character table

568 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 21080);