C₁₀₃×#7592 #39912867

order	26368
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₀₃×#7592

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₃
prime radical	C₂³×D₄×C₁₀₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₃ ◁ C₂²×C₄²×C₁₀₃ ◁ G

846 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 3 generators on 127 points:

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

Conjugacy classes & character table

5356 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 580);