C₇₃×#13611 #32923015

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

Aliases

direct product

C₇₃×#13611

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₃
prime radical	C₇₃×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #536×C₇₃ ◁ G

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Conjugacy classes & character table

1825 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6599);