C₇₃×#7375 #32916779

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7375

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂×C₈×C₇₃
Frattini, cosocle	#390	C₂²×C₇₃
prime radical	C₂²×D₄×C₇₃	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #390×C₇₃ ◁ G

228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 2336

A minimal faithful permutation representation — 2 generators on 97 points:

(1,2,3,5,4,6,7,8)(9,10)(11,18,19,14)(12,16,21,15)(13,17)(20,23,24,22)(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);(9,11,12,19,13,20,21,24)(10,14,16,18,17,22,15,23)

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, 363);