#27187909

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)(3,11)(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)(103,111)(105,117)(109,123)(112,120)(113,124)(114,128)(118,119)(129,131);(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)(102,105,107,119,108,117,106,118)(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);(2,3,4,6)(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)(102,103,107,112,108,111,106,120)(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^10,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, 21287);