C₈₉×#10574 #36388422

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

Aliases

direct product

C₈₉×#10574

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

494 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 22784

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

(1,2,5,7)(3,8,4,6)(9,10,12,15)(11,16,13,14)(19,24)(21,23)(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);(1,3,5,4)(2,6,7,8)(9,11,12,13)(10,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,3)(2,6,7,8)(10,14)(11,13)(15,16)(18,21)(19,24)(22,23)

Conjugacy classes & character table

4094 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3562);