C₇₃×#11425 #32920829

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₃×#11425

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#99
commutator	C₂³	C₂²×C₈×C₇₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×#99
prime radical	C₂⁵×C₇₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#117×C₇₃ ◁ G

678 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 97
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 97 points:

(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,19,20,14)(13,16)(17,23,24,21)(18,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(10,14)(15,21)(16,19)(22,23);(9,11,13,18)(10,15,16,22)(12,17,20,24)(14,21,19,23)

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, 4413);