C₇₃×#13051 #32922455

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₃×#13051

Subgroups

	subgroup	G/H
center	C₄×C₇₃	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×#41	C₂³×C₇₃
socle	C₂×C₇₃	#1889
prime radical	C₇₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×#41×C₇₃ ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 105
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6039);