C₈₉×#8554 #36386402

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₈₉×#8554

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#379
commutator	C₂⁴	C₂²×C₄×C₈₉
Frattini, cosocle	C₂³×C₄	C₂³×C₈₉
prime radical	C₂⁴×C₈₉	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

778 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Conjugacy classes & character table

3560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 1542);