#27173441

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃₂×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #2262

Subgroups

	subgroup	G/H
center	C₄	D₁₆₁₆
commutator, cosocle	C₁₆×C₁₀₁	C₂³
Frattini	C₁₆	C₂²×D₁₀₁
Fitting	C₂×C₃₂×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₀₁
hypocenter	C₁₀₁	#2262
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

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:

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

Polycyclic presentation:

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

3238 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 6819);