C₁₀₉×#12698 #42130692

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₀₉×#12698

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×D₄	C₂³×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

990 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 145
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 145 points:

(1,2)(3,4)(5,6)(7,15,18,30)(8,12)(9,22,24,13)(10,14)(11,17)(16,31,19,33)(20,35,36,28)(21,29)(23,26)(25,32)(27,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(5,7,10,18)(6,11,14,26)(8,19,21,16)(9,23,24,17)(12,27,29,25)(13,30,22,15)(20,32,36,34)(28,31,35,33);(2,4)(5,8,10,21)(6,12,14,29)(7,16,18,19)(9,20,24,36)(11,25,26,27)(13,28,22,35)(15,31,30,33)(17,32,23,34)

Conjugacy classes & character table

3706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5686);