C₈₉×#10534 #36388382

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#10534

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂³×C₈₉	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

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

normal subgroups only

This group has 358 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) = 113
transitive degree	μ_t(G) = 11392

A minimal faithful permutation representation — 3 generators on 113 points:

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

Conjugacy classes & character table

3026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3522);