#27187904

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	AGL(1,ℤ/8)×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #2955

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×#87938
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₄	C₂³×#87938
Fitting	AGL(1,ℤ/8)×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×C₄×D₄
prime radical	#20265973	C₂
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	AGL(1,ℤ/8)	#87938
hypocenter	C₁₀₁	#2955
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 133 points:

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

Polycyclic presentation:

2^7,101|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^91,e^b=eg,h^c=h^10,h^f=h^100

Conjugacy classes & character table

319 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 21282);