#27187529

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

Aliases

No aliases for this group.

Splittings

Fitting	Q₃₂×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #2176

Subgroups

	subgroup	G/H
center	C₂	D₈×#87938
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₈	C₂²×#87938
Fitting	Q₃₂×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₄×D₈
prime radical	C₂×D₁₀₁	C₂×D₈
cosocle	C₈×D₁₀₁	C₂³
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#2176
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2,8,4)(3,6,18,14)(5,7,16,10)(9,12,32,24)(11,17,28,20)(13,15,30,26)(19,22,50,38)(21,31,42,34)(23,25,48,44)(27,29,46,40)(33,36,51,54)(35,49,58,52)(37,39,68,60)(41,47,64,56)(43,45,66,62)(53,69,72,55)(57,67,76,70)(59,61,82,78)(63,65,80,74)(71,83,86,73)(75,81,90,84)(77,79,92,88)(85,93,96,87)(89,91,98,94)(95,99,100,97)(103,109)(108,123)(110,125)(111,121)(112,129)(116,128)(124,127)(126,132);(102,104,107,115)(103,108,109,123)(105,117,120,133)(106,122,118,114)(110,126,125,132)(111,127,121,124)(112,128,129,116)(113,130,131,119);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,54,50)(42,49,57,56)(45,59,60,48)(46,47,63,62)(58,69,71,70)(61,73,72,68)(64,67,75,74)(65,77,78,66)(76,83,85,84)(79,87,86,82)(80,81,89,88)(90,93,95,94)(91,97,96,92)(98,99,101,100)(102,103)(104,112)(105,116)(106,121)(107,109)(108,113)(110,119)(111,118)(114,132)(115,129)(117,127)(120,128)(122,126)(123,131)(124,133)(125,130)

Polycyclic presentation:

2^7,101|a^2=e,b^2=g,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^91,d^b=df,f^b=fg,h^c=h^10,h^e=h^100

Conjugacy classes & character table

307 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20907);