C₇₃×#7307 #32916711

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇₃×#7307

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#342
commutator	C₂²×C₄	C₄²×C₇₃
Frattini, cosocle	C₂³×D₄	C₂²×C₇₃
prime radical	C₂³×D₄×C₇₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂⁴×C₇₃ ◁ C₂³×D₄×C₇₃ ◁ G

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

normal subgroups only

This group has 794 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) = 89
transitive degree	μ_t(G) = 1168

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2)(3,7,12,6)(4,10)(5,9)(8,13)(11,14,16,15)(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);(2,6,8,14)(3,11)(7,9,15,10)(12,16)

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, 295);