C₈₉×#29231 #36407079

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈₉×#29231

Subgroups

	subgroup	G/H
center	C₄×C₈₉	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
socle	C₂×C₈₉	#2571
prime radical	C₂³×C₈₉	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

404 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 4 generators on 121 points:

(1,2,7,12)(3,13)(4,18,19,23)(5,8,6,11)(9,14)(10,21,26,20)(15,29)(16,24)(17,22)(25,31)(27,28)(30,32)(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);(2,8,12,11)(3,14,17,16)(4,19)(9,24)(10,18,26,23)(13,22)(15,30,31,28)(20,21);(1,3)(2,9,12,24)(4,15)(5,14)(6,16)(7,17)(8,22,11,13)(10,25,26,29)(18,27,23,32)(19,31)(20,28)(21,30);(1,4,5,20,7,19,6,21)(2,10,11,18,12,26,8,23)(3,15,14,28,17,31,16,30)(9,25,13,27,24,29,22,32)

Conjugacy classes & character table

3560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 22219);