C₈₉×#10428 #36388276

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

Aliases

direct product

C₈₉×#10428

Subgroups

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

386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 125
transitive degree	μ_t(G) = 22784

A minimal faithful permutation representation — 3 generators on 125 points:

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

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, 3416);