C₁₀₁×#1382 #27166736

order	12928
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1382

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₁	D₈
commutator	C₈	C₄²×C₁₀₁
Frattini, cosocle	C₂²×C₈	C₂²×C₁₀₁
socle	C₂²×C₁₀₁	#102
prime radical	C₂³×C₁₀₁	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂²×C₈×C₁₀₁ ◁ G

132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 137
transitive degree	μ_t(G) = 6464

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

(1,2,3,4)(5,6)(7,16)(8,21)(9,24)(10,26)(11,15)(12,17)(13,23)(14,22)(18,28)(19,36)(20,32)(25,29)(27,35)(30,34)(31,33)(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);(5,7,9,18,22,33,23,34,11,20,8,17,10,19,25,35)(6,12,13,16,26,30,24,36,15,28,29,32,14,27,21,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₁×#1382, C₁₀₁×#12561, C₄×#359×C₁₀₁, …
maximal normal subgroup of	C₂×C₁₀₁×#1382, C₁₀₁×#12561, C₄×#359×C₁₀₁, …
maximal quotient of	C₁₀₁×#7155, C₁₀₁×#7171, C₁₀₁×#7172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 114);