C₇₃×#12331 #32921735

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#12331

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₇₃×#2206	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈×Q₈×C₇₃ ◁ G

328 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 9344

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

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5319);