C₈₉×#10647 #36388495

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#10647

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#397
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

258 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

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, 3635);