C₁₀₉×#13599 #42131593

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#13599

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	C₂×#354
commutator	C₄²	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄⋊Q₈×C₁₀₉ ◁ G

312 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Conjugacy classes & character table

3052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 6587);