C₈₉×#12329 #36390177

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

Aliases

direct product

C₈₉×#12329

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₈₉
prime radical	C₈₉×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ C₈×Q₈×C₈₉ ◁ G

340 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 113
transitive degree	μ_t(G) = 11392

A minimal faithful permutation representation — 3 generators on 113 points:

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

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, 5317);