C₁₀₁×#2901 #27168255

order	12928
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₀₁×#2901

Subgroups

	subgroup	G/H
center	C₄×C₁₀₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
socle	C₂×C₁₀₁	C₂²×C₂²⋊C₄
prime radical	#585×C₁₀₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₁ ◁ C₂×#41×C₁₀₁ ◁ G

738 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 117 points:

(1,2,5,9,6,3,10,8)(4,7,13,15,12,11,16,14)(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)(5,8)(7,12)(9,10)(13,14)(15,16);(1,4)(2,7)(3,11)(5,13)(6,12)(8,14)(9,15)(10,16);(4,12)(7,11)(13,16)(14,15)

Conjugacy classes & character table

4444 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#22461, C₁₀₁×#22462, C₁₀₁×#22898, …
maximal normal subgroup of	C₁₀₁×#22461, C₁₀₁×#22462, C₁₀₁×#22898, …
maximal quotient of	C₁₀₁×#17438, C₁₀₁×#17439, C₁₀₁×#17453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1633);