C₇₃×#7078 #32916482

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

Aliases

direct product

C₇₃×#7078

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#327
commutator	C₂²×C₈	C₂×C₄×C₇₃
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₇₃
prime radical	C₂⁴×C₇₃	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₂²⋊C₈×C₇₃ ◁ G

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 4672

A minimal faithful permutation representation — 2 generators on 97 points:

(1,2)(3,7,8,5)(4,6)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22)(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);(2,5)(6,7)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17)

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, 66);