C₈₉×#10217 #36388065

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

Aliases

direct product

C₈₉×#10217

Subgroups

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

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 5696

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

(1,2)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(17,18,23,28)(19,29,32,43)(20,33,36,41)(21,27,22,26)(24,37,39,31)(25,34,42,35)(30,38,46,40)(44,47,48,45)(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);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,19,23,32)(18,24,28,39)(20,30,36,46)(21,37,22,31)(25,40,42,38)(26,43,27,29)(33,44,41,48)(34,47,35,45);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,20,23,36)(18,25,28,42)(19,30,32,46)(21,35,22,34)(24,38,39,40)(26,33,27,41)(29,44,43,48)(31,47,37,45)

Conjugacy classes & character table

2492 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3205);