C₇₃×#13605 #32923009

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#13605

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	#124	C₂³×C₇₃
prime radical	C₇₃×#3220	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #535×C₇₃ ◁ G

344 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,11,13,26)(4,15,16,31)(5,18,20,9)(6,10)(7,12,22,29)(8,17)(14,24,19,27)(21,28,23,30)(25,32)(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)(2,7)(4,12)(5,19)(6,14)(8,24)(9,22)(10,21)(11,25)(13,20)(15,27)(16,28)(17,29)(18,23)(26,31)(30,32);(4,16)(7,21)(8,25)(12,28)(15,31)(17,32)(22,23)(29,30)

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, 6593);