C₇₃×#8754 #32918158

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

Aliases

direct product

C₇₃×#8754

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#381
commutator	C₂⁴	C₂²×C₄×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
prime radical	C₂⁴×C₇₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

566 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 113
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 113 points:

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

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, 1742);