#27185278

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	#563×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3531

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#563×C₁₀₁	C₂
hypercenter	#563	D₁₀₁
hypocenter	C₁₀₁	#3531
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #563

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 5 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)(118,121,122,129)(119,125,124,131)(120,128,127,133)(123,130,126,132);(103,109)(107,116)(108,112)(115,117)(119,124)(120,127)(125,131)(128,133);(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)(102,105,106,114)(103,108,109,112)(104,110,111,113)(107,115,116,117)(120,127)(123,126)(128,133)(130,132);(102,104,106,111)(103,107,109,116)(105,113,114,110)(108,117,112,115)(118,120)(119,123)(121,128)(122,127)(124,126)(125,130)(129,133)(131,132);(102,103)(104,107)(105,112)(106,109)(108,114)(110,117)(111,116)(113,115)(118,119)(120,126)(121,125)(122,124)(123,127)(128,132)(129,131)(130,133)

Polycyclic presentation:

2^7,101|b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,h^c=h^100,h^e=h^100

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 18656);