C₁₀₁×#2249 #27167603

order	12928
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₁×#2249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₁
prime radical	C₂×D₁₆×C₁₀₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ C₄×C₈×C₁₀₁ ◁ G

184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 137 points:

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

Conjugacy classes & character table

3838 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#7446, C₁₀₁×#7447, C₁₀₁×#13689, …
maximal normal subgroup of	C₁₀₁×#7446, C₁₀₁×#7447, C₁₀₁×#13689, …
maximal quotient of	C₁₀₁×#12503, C₁₀₁×#12504, C₁₀₁×#12505, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 981);