C₂×C₁₀₉×#1655 #42136972

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

Aliases

direct product

C₂×C₁₀₉×#1655

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

1330 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 127
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 4 generators on 127 points:

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

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, 11966);