C₈₉×#10081 #36387929

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₈₉×#10081

Subgroups

	subgroup	G/H
center, socle	C₂³×C₈₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₈₉×#2819	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

648 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 117
transitive degree	μ_t(G) = 11392

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

3026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 3069);