C₇₃×#13667 #32923071

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#13667

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ C₂×M₃₂×C₇₃ ◁ G

294 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Conjugacy classes & character table

2263 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6655);