C₁₀₉×#12975 #42130969

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#12975

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×#41	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1867
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

4360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5963);