C₈₉×#12413 #36390261

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

Aliases

direct product

C₈₉×#12413

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

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

normal subgroups only

This group has 266 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) = 137
transitive degree	μ_t(G) = 22784

A minimal faithful permutation representation — 3 generators on 137 points:

(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(17,18,22,26)(19,27,29,46)(20,31,33,45)(21,36,34,25)(23,38,39,28)(24,41,43,35)(30,47,37,42)(32,44,48,40)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19,22,29)(18,23,26,39)(20,32,33,48)(21,37,34,30)(24,42,43,47)(25,44,36,40)(27,45,46,31)(28,41,38,35);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(20,33)(24,43)(27,46)(28,38)(31,45)(32,48)(35,41)(42,47)

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, 5401);