C₇₃×#8860 #32918264

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#8860

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
socle	C₂²×C₇₃	#385
prime radical	C₂²×D₄×C₇₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

458 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 97
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 97 points:

(5,8)(9,10,13,16,14,17,12,21)(11,18,19,15,20,23,22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2)(3,5)(4,6)(7,8)(9,11,13,19,14,20,12,22)(10,15,16,23,17,24,21,18);(1,3,4,7)(2,5,6,8)

Conjugacy classes & character table

3358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1848);