C₈₉×#32934 #36410782

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₈₉×#32934

Subgroups

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

1378 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 310 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 — 4 generators on 113 points:

(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,20,24)(12,21,18,16)(14,22,23,19)(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);(9,11,18,22,13,20,12,19)(10,14,21,24,15,23,16,17);(2,4)(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24);(6,8)(10,15)(14,23)(16,21)(17,24)

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, 25922);