C₇₃×#11652 #32921056

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#11652

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₇₃×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

344 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Conjugacy classes & character table

3358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4640);