#27169073

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1323

Subgroups

	subgroup	G/H
center	C₂×C₄	#13108051
commutator	C₂×C₄×C₁₀₁	C₂×C₈
Frattini	C₂²×C₈	C₂×D₁₀₁
Fitting	C₂²⋊C₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²⋊C₈
prime radical	C₂³×C₁₀₁	M₁₆
cosocle	C₂²×C₈×C₁₀₁	C₂²
hypercenter	C₂²⋊C₁₆	D₁₀₁
hypocenter	C₁₀₁	#1323
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 2 generators on 141 points:

(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)(102,104,105,109)(103,106,107,108)(110,112)(111,117)(113,126)(114,128)(115,123)(116,125)(118,134)(119,136)(120,122)(121,133)(124,130)(127,140)(129,131)(132,138)(135,139)(137,141);(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)(102,103)(104,108,109,106)(105,107)(110,111,114,119,115,120,129,137,116,121,130,138,113,118,127,135)(112,122,124,139,123,133,140,136,125,134,128,141,126,117,131,132)

Polycyclic presentation:

2^7,101|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,c^a=ce,c^b=cg,d^b=deg,h^b=h^100

Conjugacy classes & character table

2038 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 2451);