C₁₀₉×#12779 #42130773

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

Aliases

direct product

C₁₀₉×#12779

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1790
commutator	C₂×Q₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×#41	C₂³×C₁₀₉
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

732 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,13,16,27)(4,10,5,18)(6,11)(7,12)(8,14)(9,17,26,20)(15,29,32,22)(19,24)(21,25)(23,30)(28,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,7,16)(2,8,12,24)(4,14,5,19)(6,15,21,32)(9,22,26,29)(10,27,18,13)(11,28,25,23)(17,30,20,31);(1,4)(2,9)(3,14)(5,7)(6,20)(8,22)(10,25)(11,18)(12,26)(13,28)(15,31)(16,19)(17,21)(23,27)(24,29)(30,32)

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, 5767);