C₈₉×#8543 #36386391

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₈₉×#8543

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#379
commutator	C₂⁴	C₂²×C₄×C₈₉
Frattini, cosocle	C₂⁵	C₂³×C₈₉
prime radical	C₂³×D₄×C₈₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂⁵×C₈₉ ◁ G

1266 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 2848

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

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

Conjugacy classes & character table

3560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 1531);