C₈₉×#7478 #36385326

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#7478

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#357
commutator	C₁₆	C₄²×C₈₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₈₉
prime radical	C₂³×C₈₉	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₈×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

174 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 11392

A minimal faithful permutation representation — 2 generators on 125 points:

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

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, 466);