C₈₉×#7352 #36385200

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₈₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₉×#7352

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#353
commutator	C₄²	C₂×C₈×C₈₉
Frattini, cosocle	#382	C₂²×C₈₉
prime radical	C₂×#127×C₈₉	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #382×C₈₉ ◁ G

256 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 113 points:

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

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, 340);