C₂×C₇₃×#2201 #32942953

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₇₃×#2201

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁴×C₇₃
prime radical	C₂×H(5,𝔽₂)×C₇₃	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₇₃ ◁ C₂²×D₄×C₇₃ ◁ G

1136 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 107
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 4 generators on 107 points:

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

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, 26537);