C₈₉×#7336 #36385184

order	22784
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#7336

Subgroups

	subgroup	G/H
center	C₈×C₈₉	#99
commutator	C₂³	C₂×C₁₆×C₈₉
Frattini, cosocle	C₂×M₃₂	C₂²×C₈₉
socle	C₂×C₈₉	#1314
prime radical	C₂⁴×C₈₉	C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₈₉ ◁ C₂×C₈×C₈₉ ◁ C₂×M₃₂×C₈₉ ◁ G

170 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 121 points:

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

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, 324);