C₈₉×#9758 #36387606

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₈₉×#9758

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#390
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

408 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 129
transitive degree	μ_t(G) = 5696

A minimal faithful permutation representation — 3 generators on 129 points:

(2,5)(3,7)(6,8)(9,10)(11,15,20,30)(12,22)(13,26)(14,17)(16,24)(18,27)(19,33,23,37)(21,36,35,29)(25,38)(28,39,31,34)(32,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,2,3,5,4,6,7,8)(9,11)(10,15,17,30)(12,23)(13,21)(14,20)(16,31,32,28)(18,29,26,36)(19,25)(22,37,38,33)(24,39)(27,35)(34,40);(2,6)(5,8)(9,12,13,24,14,25,27,40)(10,16,18,22,17,32,26,38)(11,19,21,34,20,23,35,39)(15,28,29,33,30,31,36,37)

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, 2746);