C₇₃×#8242 #32917646

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#8242

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₇₃	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

384 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 105
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

5548 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1230);