C₈₉×#10097 #36387945

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

Aliases

direct product

C₈₉×#10097

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

542 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,21,25)(19,23,27,32)(20,22,28,24)(26,31,29,30)(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);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(18,22)(20,28)(23,31)(24,25)(26,29)(30,32);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,19,21,27)(18,23,25,32)(20,29,28,26)(22,30,24,31)

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, 3085);