C₈₉×#10319 #36388167

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈₉×#10319

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 125
transitive degree	μ_t(G) = 22784

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2)(3,4)(6,11)(8,9)(12,15)(13,14)(16,17)(19,20)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,33,36,32)(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);(2,4)(5,6,10,14)(7,15,18,19)(8,13,9,11)(12,16,20,17)(21,23,26,34)(22,27,30,36)(24,33,25,32)(28,35,29,31);(5,7,10,18)(6,12,14,20)(8,17,9,16)(11,15,13,19)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33)

Conjugacy classes & character table

4628 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3307);