C₁₀₉×#32705 #42150699

order	27904
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₉×#32705

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂×C₁₀₉	#2804
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

536 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 141 points:

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

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, 25693);