C₇₃×#12309 #32921713

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#12309

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₃
prime radical	#469×C₇₃	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×M₁₆×C₇₃ ◁ G

280 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 2336

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

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

Conjugacy classes & character table

1825 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5297);