C₁₀₁×#1817 #27167171

order	12928
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₁×#1817

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂³
commutator	C₂²	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

380 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 6464

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7017, C₁₀₁×#7021, C₁₀₁×#7033, …
maximal normal subgroup of	C₁₀₁×#7017, C₁₀₁×#7021, C₁₀₁×#7033, …
maximal quotient of	C₁₀₁×#8698, C₁₀₁×#8699, C₁₀₁×#8700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 549);