#27182908

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#444×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3289

Subgroups

	subgroup	G/H
center	C₂	D₄×D₄₀₄
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#444×C₁₀₁	C₂
socle	C₂×C₁₀₁	D₄²
hypercenter	#444	D₁₀₁
hypocenter	C₁₀₁	#3289
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #444

448 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:

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

Polycyclic presentation:

2^7,101|a^2=g,b^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^100,d^b=de,e^b=eg,f^c=fg,h^c=h^100,e^d=eg

Conjugacy classes & character table

1126 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 16286);