#27185269

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

Aliases

No aliases for this group.

Splittings

Fitting	#562×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3529

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#562×C₁₀₁	C₂
hypercenter	#562	D₁₀₁
hypocenter	C₁₀₁	#3529
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #562

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 5 generators on 125 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)(110,113,114,121)(111,117,116,123)(112,120,119,125)(115,122,118,124);(103,107)(106,109)(111,116)(112,119)(117,123)(120,125);(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)(103,107)(104,108)(112,119)(115,118)(120,125)(122,124);(102,104)(103,106)(105,108)(107,109)(110,112)(111,115)(113,120)(114,119)(116,118)(117,122)(121,125)(123,124);(102,103)(104,106)(105,107)(108,109)(110,111)(112,118)(113,117)(114,116)(115,119)(120,124)(121,123)(122,125)

Polycyclic presentation:

2^7,101|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, 18647);