C₈₉×#12415 #36390263

order	22784
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#12415

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₈₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₈₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ #446×C₈₉ ◁ G

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

normal subgroups only

This group has 272 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) = 22784

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

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

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, 5403);