C₁₀₁×#1552 #27166906

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1552

Subgroups

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

208 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 125
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 125 points:

(2,6)(3,4)(7,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,2,5,7)(3,6,4,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,3,5,4)(2,6,7,8)(10,15)(14,17)(16,23)(22,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#8457, C₁₀₁×#8458, C₁₀₁×#8460, …
maximal normal subgroup of	C₁₀₁×#8457, C₁₀₁×#8458, C₁₀₁×#8460, …
maximal quotient of	C₁₀₁×#7579, C₁₀₁×#7585, C₁₀₁×#7613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 284);