C₁₀₁×#1772 #27167126

order	12928
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₁×#1772

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₀₁	D₄
commutator	C₄	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂²×C₁₀₁	C₄×D₄
prime radical	C₂³×C₁₀₁	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₁ ◁ C₂²×C₈×C₁₀₁ ◁ G

232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

5656 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#7179, C₁₀₁×#7180, C₁₀₁×#7181, …
maximal normal subgroup of	C₁₀₁×#7179, C₁₀₁×#7180, C₁₀₁×#7181, …
maximal quotient of	C₁₀₁×#8266, C₁₀₁×#8267, C₁₀₁×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 504);