C₇₃×#12710 #32922114

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₃×#12710

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1781
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₃
prime radical	C₇₃×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂³×C₇₃ ◁ C₂³×Q₈×C₇₃ ◁ G

952 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5698);