C₁₀₃×#7109 #39912384

order	26368
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₃×#7109

Subgroups

	subgroup	G/H
center	C₄×C₁₀₃	#327
commutator	C₂×C₈	C₂×C₈×C₁₀₃
Frattini, cosocle	C₂×#97	C₂²×C₁₀₃
socle	C₂×C₁₀₃	#1270
prime radical	C₂×#135×C₁₀₃	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₃ ◁ C₂²×C₄×C₁₀₃ ◁ C₂×#97×C₁₀₃ ◁ G

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 3296

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

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

Conjugacy classes & character table

4120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 97);