C₇₃×#10535 #32919939

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#10535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

342 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 105
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3523);