#27185902

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	#578×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3584

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₁
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	#578×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂³×D₄
hypercenter	#578	D₁₀₁
hypocenter	C₁₀₁	#3584
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^7,101|e^2=g,f^2=g,b^a=bf,f^a=fg,f^b=fg,d^c=dg,h^c=h^100,h^d=h^100,h^e=h^100

Conjugacy classes & character table

1144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19280);