C₈₉×#7523 #36385371

order	22784
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#7523

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1404
commutator	#125	C₂×C₄×C₈₉
Frattini, cosocle	#490	C₂²×C₈₉
prime radical	C₈₉×#1656	C₂
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₄²×C₈₉ ◁ #490×C₈₉ ◁ G

158 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 105
transitive degree	μ_t(G) = 1424

A minimal faithful permutation representation — 2 generators on 105 points:

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

Conjugacy classes & character table

1691 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 511);