C₂×#464×C₁₀₁ #27168388

order	12928
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#464×C₁₀₁

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
prime radical	C₂²×D₄×C₁₀₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 4 generators on 123 points:

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

Conjugacy classes & character table

3838 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#9244, C₄×#464×C₁₀₁, C₁₀₁×#21508, …
maximal normal subgroup of	C₁₀₁×#9244, C₄×#464×C₁₀₁, C₁₀₁×#21508, …
maximal quotient of	C₂×C₁₀₁×#1678, C₂×C₁₀₁×#1680, C₂×C₁₀₁×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1766);

C2×#464×C101 #27168388