C₇₃×#7396 #32916800

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7396

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#353
commutator	C₂×M₁₆	C₂×C₄×C₇₃
Frattini, cosocle	#407	C₂²×C₇₃
socle	C₂×C₇₃	#1343
prime radical	C₇₃×#2010	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #407×C₇₃ ◁ G

296 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

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, 384);