C₁₀₃×#10381 #39915656

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

Aliases

direct product

C₁₀₃×#10381

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₃
prime radical	C₂³×C₁₀₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂×C₄²×C₁₀₃ ◁ G

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

normal subgroups only

This group has 352 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:

(1,2,4,7)(3,10)(5,9,6,14)(8,11)(12,15)(13,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31)(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);(2,7)(3,11,13,12)(8,16,15,10)(9,14)(20,29)(22,31)(23,28)(26,30);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25)

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, 3369);