C₇₃×#7534 #32916938

order	18688
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₃×#7534

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1409
commutator	C₄×C₈	C₂×C₄×C₇₃
Frattini, cosocle	#494	C₂²×C₇₃
prime radical	C₂²×C₇₃	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄²×C₇₃ ◁ #494×C₇₃ ◁ G

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 2336

A minimal faithful permutation representation — 2 generators on 105 points:

(1,2,4,10,7,12,20,27)(3,9,14,25,16,23,30,32)(5,11,19,28,6,21,18,26)(8,17,22,13,24,29,31,15)(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);(2,8,12,24)(3,13)(5,6)(9,21,23,11)(10,25,27,32)(14,29)(15,16)(17,30)(18,19)(22,28,31,26)

Conjugacy classes & character table

1387 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 522);