C₂²×#376×C₁₀₉ #42151981

order	27904
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂²×#376×C₁₀₉

Subgroups

	subgroup	G/H
center	C₂⁴×C₄×C₁₀₉	C₂²
commutator	C₂	C₂³×C₄²×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
socle, prime radical	C₂⁵×C₁₀₉	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₁₀₉ ◁ G

5150 conjugacy classes of subgroups; only the 3614 normal subgroups are shown.

normal subgroups only

This group has 5150 conjugacy classes of subgroups — too many to draw. Only its 3614 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) = 129
transitive degree	μ_t(G) = 27904

A minimal faithful permutation representation — 5 generators on 129 points:

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

Conjugacy classes & character table

17440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 26975);