#27187911

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#135×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ #2957

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×#87938
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₄	C₂³×#87938
Fitting	#135×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×C₄×D₄
prime radical	#135×D₁₀₁	C₂
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	#135	#87938
hypocenter	C₁₀₁	#2957
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 133 points:

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

Polycyclic presentation:

2^7,101|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^91,e^b=eg,d^c=dg,h^c=h^10,h^d=h^100,h^f=h^100

Conjugacy classes & character table

394 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 21289);