C₁₀₃×#10389 #39915664

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

Aliases

direct product

C₁₀₃×#10389

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₃
prime radical	C₂⁴×C₁₀₃	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂³×C₄×C₁₀₃ ◁ G

406 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 135
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2,6,9)(3,10)(4,8,5,14)(7,11)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(3,11,12,13)(7,10,15,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30)

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, 3377);