C₂×C₁₀₁×#1324 #38793596

order	25856
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₁₀₁×#1324

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₁	#99
commutator	C₂×C₄	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂⁴×C₁₀₁ ◁ C₂²×C₄⋊C₄×C₁₀₁ ◁ G

510 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 127
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 127 points:

(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(3,4,5,7,6,8,9,10)(11,12)(13,19,21,25)(14,22,23,17)(15,18)(16,20)(24,26);(11,13)(12,16,18,24)(14,20)(15,21)(17,19,22,25)(23,26)

Conjugacy classes & character table

5252 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25856, 4433);