C₈₉×#13598 #36391446

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈₉×#13598

Subgroups

	subgroup	G/H
center	C₄×C₈₉	#354
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₈₉
socle	C₂×C₈₉	C₂×#354
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #532×C₈₉ ◁ G

318 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

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, 6586);