C₈₉×#10041 #36387889

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

Aliases

direct product

C₈₉×#10041

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

478 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 121
transitive degree	μ_t(G) = 11392

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

2759 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3029);