C₈₉×#9446 #36387294

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₈₉×#9446

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂³×C₄	C₂³×C₈₉
prime radical	C₂×#121×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 5696

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

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

Conjugacy classes & character table

3293 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 2434);