C₇₃×#13683 #32923087

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

Aliases

direct product

C₇₃×#13683

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₃
prime radical	C₇₃×#3291	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄⋊Q₈×C₇₃ ◁ G

392 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 89
transitive degree	μ_t(G) = 1168

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,7,6)(3,11,14,16)(4,15,10,5)(8,13,9,12)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,3)(2,8)(4,13)(5,16)(6,9)(7,14)(10,12)(11,15);(1,3)(2,9)(4,11)(5,12)(6,8)(7,14)(10,16)(13,15)

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, 6671);