C₇₃×#11764 #32921168

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

Aliases

direct product

C₇₃×#11764

Subgroups

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

446 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 97 points:

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

Conjugacy classes & character table

2701 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4752);