C₁₀₃×#7092 #39912367

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

Aliases

direct product

C₁₀₃×#7092

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#331
commutator	C₄×C₈	C₂×C₄×C₁₀₃
Frattini, cosocle	C₈²	C₂²×C₁₀₃
prime radical	#493×C₁₀₃	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₄²×C₁₀₃ ◁ C₈²×C₁₀₃ ◁ G

152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 13184

A minimal faithful permutation representation — 2 generators on 135 points:

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

Conjugacy classes & character table

4429 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 80);