#27187556

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #2258

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×#87938
commutator, socle	C₂×C₁₀₁	C₂²×C₁₆
Fitting	M₃₂×C₁₀₁	C₄
prime radical	#13108074	C₈
cosocle	#13108049	C₂³
hypercenter	M₃₂	#87938
hypocenter	C₁₀₁	#2258
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₈ ◁ M₃₂

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 165 points:

(1,2,5,10,19,32,49,42,25,12,3,7,14,23,36,53,68,59,38,21,8,16,27,40,57,72,83,74,55,34,17,29,44,61,76,87,94,85,70,51,30,46,64,79,90,97,99,92,81,66,47,65,80,91,98,101,100,96,89,78,63,45,62,77,88,95,93,86,75,60,43,28,41,58,73,84,82,71,56,39,26,15,24,37,54,69,67,52,35,22,13,6,11,20,33,50,48,31,18,9,4)(104,114)(109,124)(115,134)(116,135)(117,136)(125,144)(126,145)(127,146)(137,154)(138,155)(139,156)(147,159)(148,160)(149,161)(157,164)(162,165);(2,4)(3,6)(5,9)(7,13)(8,15)(10,18)(11,12)(14,22)(16,26)(17,28)(19,31)(20,25)(21,24)(23,35)(27,39)(29,43)(30,45)(32,48)(33,42)(34,41)(36,52)(37,38)(40,56)(44,60)(46,63)(47,65)(49,50)(51,62)(53,67)(54,59)(55,58)(57,71)(61,75)(64,78)(66,80)(68,69)(70,77)(72,82)(73,74)(76,86)(79,89)(81,91)(83,84)(85,88)(87,93)(90,96)(92,98)(94,95)(97,100)(99,101)(102,104)(103,109)(105,115)(106,116)(107,117)(108,114)(110,125)(111,126)(112,127)(113,124)(118,137)(119,138)(120,134)(121,139)(122,135)(123,136)(128,147)(129,148)(130,144)(131,149)(132,145)(133,146)(140,157)(141,154)(142,155)(143,156)(150,162)(151,159)(152,160)(153,161)(158,164)(163,165);(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)(102,103,105,110,106,111,118,128,107,112,119,129,121,131,140,150,108,113,120,130,122,132,141,151,123,133,142,152,143,153,158,163)(104,109,115,125,116,126,137,147,117,127,138,148,139,149,157,162,114,124,134,144,135,145,154,159,136,146,155,160,156,161,164,165)

Polycyclic presentation:

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

Conjugacy classes & character table

580 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20934);