C₇₃×#11668 #32921072

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#11668

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₂³×C₇₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

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

normal subgroups only

This group has 354 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) = 113
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 113 points:

(1,2,3,6,4,5,7,8)(9,10)(11,20,23,36)(12,16)(13,27)(14,19)(15,22,31,39)(17,25)(18,26)(21,34,24,37)(28,30,38,32)(29,33)(35,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(9,11,14,23)(10,15,19,31)(12,24,26,21)(13,28,29,38)(16,32,18,30)(17,34,35,37)(20,27,36,33)(22,25,39,40);(2,5)(6,8)(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,40)(15,30,31,32)(17,33,35,27)(20,34,36,37)(22,38,39,28)

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, 4656);