#27186280

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₈×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #1414

Subgroups

	subgroup	G/H
center	C₂	#20266084
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	C₄×C₈×C₁₀₁	C₄
socle	C₂×C₁₀₁	#354
prime radical	C₂×D₄₀₄	D₄
cosocle	#15692243	C₂²
hypercenter	C₄×C₈	#87938
hypocenter	C₁₀₁	#1414
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 133 points:

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

Polycyclic presentation:

2^7,101|a^2=d,b^2=f,c^2=f,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^91,c^b=cfg,d^b=defg,e^b=eg,h^b=h^100,d^c=dfg,e^d=eg,h^d=h^100

Conjugacy classes & character table

817 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19658);