#27170741

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

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1710

Subgroups

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#500×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#127
hypercenter	#500	D₁₀₁
hypocenter	C₁₀₁	#1710
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 12928

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

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

Polycyclic presentation:

2^7,101|a^2=f,b^2=dg,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^b=h^100,h^c=h^100

Conjugacy classes & character table

1138 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 4119);