C₄×#353×C₇₃ #32922029

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₄×#353×C₇₃

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#99
commutator	C₂×C₄	C₂×C₄²×C₇₃
Frattini, cosocle	C₂²×D₄	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×#99
prime radical	C₂×#127×C₇₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₄×D₄×C₇₃ ◁ G

720 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 85
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 85 points:

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

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, 5613);