#27172262

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#99×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #2022

Subgroups

	subgroup	G/H
center	C₂²	#15692225
commutator, cosocle	C₂⁴×C₁₀₁	C₂³
Frattini	C₂⁴	C₂²×D₁₀₁
Fitting	C₂×#99×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	#20265736	C₂
hypercenter	C₂×#99	D₁₀₁
hypocenter	C₁₀₁	#2022
LCS	G ▷ C₂⁴×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#99

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 121 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)(103,105)(106,109)(107,112)(108,114)(110,116)(111,117)(113,120)(115,121)(118,119);(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)(106,108,110,115)(107,111,113,118)(109,114,116,121)(112,119,120,117);(102,103)(104,105)(106,107)(108,111)(109,112)(110,113)(114,117)(115,118)(116,120)(119,121)

Polycyclic presentation:

2^7,101|b^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,h^b=h^100,d^c=dg,h^c=h^100

Conjugacy classes & character table

1126 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 5640);