C₈₉×#9506 #36387354

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₈₉×#9506

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₂⁴	C₂²×C₄×C₈₉
Frattini, cosocle	C₂³×C₄	C₂³×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

682 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 5696

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

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

Conjugacy classes & character table

3293 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 2494);