C₂×#357×C₁₀₁ #27167490

order	12928
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#357×C₁₀₁

Subgroups

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

272 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 123
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 123 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×#357×C₁₀₁, C₁₀₁×#12637, C₁₀₁×#12731, …
maximal normal subgroup of	C₄×#357×C₁₀₁, C₁₀₁×#12637, C₁₀₁×#12731, …
maximal quotient of	C₂×C₁₀₁×#1331, C₁₀₁×#11645, C₁₀₁×#11657, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 868);