C₂³×M₁₆×C₁₀₁ #27168924

order	12928
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂³×M₁₆×C₁₀₁

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₁₀₁	C₂²
commutator	C₂	C₂⁴×C₄×C₁₀₁
Frattini, cosocle	C₄	C₂⁵×C₁₀₁
socle	C₂⁴×C₁₀₁	C₂×C₄
prime radical	C₂⁵×C₁₀₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

1720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 5 generators on 115 points:

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

Conjugacy classes & character table

8080 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#10715, C₁₀₁×#10717, C₁₀₁×C₂²⋊#338, …
maximal normal subgroup of	C₁₀₁×#10715, C₁₀₁×#10717, C₁₀₁×C₂²⋊#338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 2302);

C23×M16×C101 #27168924