C₇₃×#9436 #32918840

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#9436

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#386
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

400 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 2424);