C₈₉×#7349 #36385197

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#7349

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1317
commutator	C₂⁴	C₂×C₈×C₈₉
Frattini, cosocle	#379	C₂²×C₈₉
prime radical	C₂⁵×C₈₉	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂³×C₈₉ ◁ C₂⁴×C₈₉ ◁ #379×C₈₉ ◁ G

302 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 46 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:

(1,2,4,8)(3,6,10,13,9,14,11,16)(5,7,12,15)(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);(1,3)(2,6,7,14)(4,11)(5,9)(8,13,15,16)(10,12)

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, 337);