C₇₃×#13600 #32923004

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#13600

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	C₂×#354
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₃
prime radical	#532×C₇₃	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #532×C₇₃ ◁ G

318 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 105 points:

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

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6588);