C₈₃×#10516 #35238340

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₃×#10516

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
prime radical	C₂³×C₈₃	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

364 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 123
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 3 generators on 123 points:

(1,2)(3,6)(4,7)(5,8)(9,10,14,18)(11,19,22,34)(12,23,26,16)(13,27,29,17)(15,28,31,21)(20,36,24,35)(25,33,40,38)(30,37,32,39)(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);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,31)(12,24,26,20)(13,28,29,21)(16,32,23,30)(17,34,27,19)(25,37,40,39)(33,35,38,36);(2,6)(3,4)(7,8)(9,12,13,25,14,26,29,40)(10,16,17,33,18,23,27,38)(11,20,21,37,22,24,28,39)(15,30,19,35,31,32,34,36)

Conjugacy classes & character table

2822 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 3504);