C₂²×#17989430 #34015856

order	18688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#17989430

Splittings

Sylow

C₇₃ ⋊ C₂²×#323

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²⋊#64408
commutator	C₂²×C₇₃	C₂³×C₈
Frattini	C₂³	C₂³×#64408
Fitting, prime radical	C₂⁶×C₇₃	C₄
socle	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	C₂⁶	#64408
hypocenter	C₇₃	C₂²×#323
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵ ◁ C₂⁶

2878 conjugacy classes of subgroups; only the 480 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 480 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) = 93
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 4 generators on 93 points:

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

Conjugacy classes & character table

1240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1099440);