C₇₃×#13053 #32922457

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

Aliases

direct product

C₇₃×#13053

Subgroups

	subgroup	G/H
center	C₄×C₇₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×#41	C₂³×C₇₃
socle	C₂×C₇₃	#1889
prime radical	C₇₃×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×#41×C₇₃ ◁ G

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

normal subgroups only

This group has 624 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)(3,12)(4,9,13,22)(5,10)(7,17)(8,20)(14,25)(15,30,19,29)(16,27)(18,26)(21,28)(23,32,24,31)(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);(1,3,13,14,6,16,4,17)(2,7,22,12,11,25,9,27)(5,19,21,32,20,15,18,31)(8,23,26,29,10,24,28,30);(2,8)(5,20)(7,23)(9,28)(10,11)(12,29)(15,19)(18,21)(22,26)(24,25)(27,30)(31,32)

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, 6041);