C₇₃×#8765 #32918169

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

Aliases

direct product

C₇₃×#8765

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#381
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂²×D₄×C₇₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

498 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 97 points:

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

Conjugacy classes & character table

3139 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1753);