#27169043

order	12928
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1307

Subgroups

	subgroup	G/H
center	C₂²	#15692162
commutator, prime radical	C₂³×C₁₀₁	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₁
Fitting	#423×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#95
cosocle	C₂×C₄²×C₁₀₁	C₂²
hypercenter	#423	D₁₀₁
hypocenter	C₁₀₁	#1307
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 2 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)(102,104,105,107,106,108,109,103)(110,112)(111,115,117,118)(113,120)(114,119)(116,123,124,125)(121,122);(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)(102,103,105,108,106,107,109,104)(110,111,113,116,114,117,121,124)(112,118,120,125,119,115,122,123)

Polycyclic presentation:

2^7,101|a^2=d,b^2=e,d^2=g,e^2=f,b^a=bc,c^a=cf,e^a=eg,h^a=h^100,c^b=cg,d^b=df,h^b=h^100

Conjugacy classes & character table

1426 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 2421);