C₂²×#17989487 #34029867

order	18688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#17989487

Splittings

Fitting	C₂⁶×C₇₃ ⋊ C₄
Sylow	C₇₃ ⋊ C₂²×AΣL(1,𝔽₁₆)

Subgroups

	subgroup	G/H
center	C₂³	#14556649
commutator	C₂³×C₇₃	C₂³×C₄
Frattini	C₂³	C₂³×#64408
Fitting	C₂⁶×C₇₃	C₄
socle	C₂³×C₇₃	#99
prime radical	C₂²×#14556560	C₂
cosocle	C₂×C₂²⋊D₇₃	C₂⁴
hypercenter	C₂⁶	#64408
hypocenter	C₇₃	C₂²×AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂⁵ ◁ C₂⁶

2774 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 2774 conjugacy classes of subgroups — too many to draw. Only its 240 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) = 85
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 4 generators on 85 points:

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

Conjugacy classes & character table

1204 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1113451);