#27186813

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	#117×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #1733

Subgroups

	subgroup	G/H
center	C₄	C₂×C₄⋊#87938
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#87938
Fitting	#117×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×#95
prime radical	C₂³×D₁₀₁	D₄
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	#117	#87938
hypocenter	C₁₀₁	#1733
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(104,111)(107,115)(110,116)(112,117);(102,104)(103,107,109,115)(105,112)(106,111)(108,110,113,116)(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)(102,103)(104,110)(105,113)(106,109)(107,112)(108,114)(111,116)(115,117)

Polycyclic presentation:

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

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20191);