#27184338

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	#520×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator	C₂×C₁₀₁	C₂⁴×C₄
Fitting	#520×C₁₀₁	C₂
socle, cosocle	C₂²×C₁₀₁	C₂⁵
prime radical	C₂×#15692301	C₂
hypercenter	#520	D₁₀₁
hypocenter	C₁₀₁	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #520

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 5 generators on 113 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)(102,104,105,103);(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)(107,111)(110,112);(108,113)(110,112);(106,108)(107,110)(109,113)(111,112);(102,103,105,104)(106,107)(108,112)(109,111)(110,113)

Polycyclic presentation:

2^7,101|a^2=g,e^2=g,b^a=bf,d^a=df,c^b=cf,h^d=h^100,h^e=h^100

Conjugacy classes & character table

1768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 17716);