C₁₀₁×#1827 #27167181

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₁×#1827

Subgroups

	subgroup	G/H
center	C₄×C₁₀₁	#117
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂×C₁₀₁	#381
prime radical	C₂×#41×C₁₀₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₁ ◁ #117×C₁₀₁ ◁ G

262 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 117
transitive degree	μ_t(G) = 1616

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#12719, C₁₀₁×#12720, C₁₀₁×#12721, …
maximal normal subgroup of	C₁₀₁×#12719, C₁₀₁×#12720, C₁₀₁×#12721, …
maximal quotient of	C₁₀₁×#8730, C₁₀₁×#8731, C₁₀₁×#8732, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 559);