C₇₃×#7253 #32916657

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

Aliases

direct product

C₇₃×#7253

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#99
commutator	C₂²×C₄	C₄²×C₇₃
Frattini, cosocle	C₂×#117	C₂²×C₇₃
socle	C₂²×C₇₃	#342
prime radical	C₂⁴×C₇₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#117×C₇₃ ◁ G

328 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 113
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 2 generators on 113 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,20,14,19)(11,21,23,38,25,34,24,39)(12,17,27,35,28,36,29,37)(16,30,31,26,33,40,32,22)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5,7,6)(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,31)(12,26,27,40,28,22,29,30)(17,34,35,39,36,21,37,38)

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, 241);