C₇₃×#7645 #32917049

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#7645

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂²×#41×C₇₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

630 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,7,8,5)(4,6)(9,10,11,14,12,13,15,16)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(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)(2,5)(4,8)(6,7)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,6)(5,7)(10,13)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Conjugacy classes & character table

3796 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 633);