C₁₀₁×#1397 #27166751

order	12928
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₁₀₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₁×#1397

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₀₁	C₂²
commutator	C₂	C₄×C₁₆×C₁₀₁
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₀₁
socle, prime radical	C₂²×C₁₀₁	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₁ ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 2 generators on 137 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7330, C₁₀₁×#7332, C₁₀₁×#7380, …
maximal normal subgroup of	C₁₀₁×#7330, C₁₀₁×#7332, C₁₀₁×#7380, …
maximal quotient of	C₁₀₁×#7329, C₁₀₁×#7331, C₁₀₁×#7465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 129);

C101×#1397 #27166751