C₈₉×#11483 #36389331

order	22784
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₈₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈₉×#11483

Subgroups

	subgroup	G/H
center	C₈×C₈₉	#99
commutator	C₂³	C₂²×C₈×C₈₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₈₉
socle	C₂×C₈₉	#1459
prime radical	#525×C₈₉	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₈₉ ◁ C₂×C₈×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

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

normal subgroups only

This group has 382 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) = 121
transitive degree	μ_t(G) = 2848

A minimal faithful permutation representation — 3 generators on 121 points:

(1,2,4,9,5,10,15,22,6,7,14,18,16,19,26,29)(3,8,11,20,12,21,24,30,13,17,23,27,25,28,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,7)(8,17)(10,19)(21,28);(1,3,5,12,6,13,16,25)(2,8,10,21,7,17,19,28)(4,11,15,24,14,23,26,31)(9,20,22,30,18,27,29,32)

Conjugacy classes & character table

4628 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 4471);