C₁₀₉×#12709 #42130703

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#12709

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

986 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 141 points:

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

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, 5697);