#27173452

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #2264

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₆₁₆
commutator, cosocle	C₁₆×C₁₀₁	C₂³
Frattini	C₁₆	C₂²×D₁₀₁
Fitting	M₆₄×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×D₁₆
prime radical	#20265609	C₂
hypercenter	M₆₄	D₁₀₁
hypocenter	C₁₀₁	#2264
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

104 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:

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

Polycyclic presentation:

2^7,101|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^100,d^b=de,e^b=ef,f^b=fg,h^b=h^100

Conjugacy classes & character table

2026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 6830);