C₈₉×#7428 #36385276

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#7428

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#354
commutator	C₂×C₄²	C₂×C₄×C₈₉
Frattini, cosocle	#390	C₂²×C₈₉
prime radical	C₂²×D₄×C₈₉	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #390×C₈₉ ◁ G

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

normal subgroups only

This group has 226 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) = 113
transitive degree	μ_t(G) = 5696

A minimal faithful permutation representation — 2 generators on 113 points:

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

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, 416);