#27186248

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1400

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊#87938
commutator	C₂²×C₁₀₁	C₂×C₁₆
Frattini	C₂×C₈	C₂×#87938
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂²⋊C₁₆
prime radical	C₂²×D₁₀₁	C₁₆
cosocle	C₂×#13108049	C₂²
hypercenter	C₂×C₁₆	#87938
hypocenter	C₁₀₁	#1400
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 133 points:

(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,80)(48,49)(51,77)(52,53)(55,73)(56,57)(59,69)(60,61)(63,64)(66,91)(67,68)(70,88)(71,72)(74,84)(75,76)(78,79)(81,98)(82,83)(85,95)(86,87)(89,90)(92,101)(93,94)(96,97)(99,100)(103,108)(104,112)(110,119)(111,120)(113,124)(114,125)(123,130)(126,132);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,65)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,49,63,50)(53,66,67,80)(57,70,71,77)(61,74,75,73)(64,68,78,69)(72,81,82,91)(76,85,86,88)(79,83,89,84)(87,92,93,98)(90,94,96,95)(97,99,100,101)(102,103,104,109,105,110,113,121,106,111,114,122,115,123,126,131,107,108,112,118,116,119,124,128,117,120,125,129,127,130,132,133)

Polycyclic presentation:

2^7,101|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc,c^a=cg,h^a=h^10,d^b=dg,h^b=h^100,h^d=h^100

Conjugacy classes & character table

844 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19626);