C₇₃×#10235 #32919639

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₃×#10235

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₇₃×#2537	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

712 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 93
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 93 points:

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

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, 3223);