C₁₀₃×#7080 #39912355

order	26368
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₃×#7080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#327
commutator	C₂²×C₈	C₂×C₄×C₁₀₃
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₁₀₃
prime radical	C₂²×D₈×C₁₀₃	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂²×C₄×C₁₀₃ ◁ C₂×C₄⋊C₈×C₁₀₃ ◁ G

240 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 6592

A minimal faithful permutation representation — 2 generators on 127 points:

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

Conjugacy classes & character table

3502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 68);