C₂×#22624855 #34009781

order	18688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#22624855

Splittings

Sylow

C₇₃ ⋊ C₂×#1316

Subgroups

	subgroup	G/H
center	C₂³	#14556649
commutator	C₂³×C₇₃	C₂²×C₈
Frattini	C₂⁴	C₂²×#64408
Fitting, prime radical	C₂⁶×C₇₃	C₄
socle	C₂³×C₇₃	#99
cosocle	C₂×C₂²⋊Dic₇₃	C₂³
hypercenter	C₂⁶	#64408
hypocenter	C₇₃	C₂×#1316
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂⁵ ◁ C₂⁶

1886 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 91
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 91 points:

(12,15)(16,17)(19,20,23,28,37,50,67,52,35,47,62,78,87,91,90,85,73,57,44,33,42,55,72,80,60,43,30,21,25,32,41,54,71,84,69,48,64,81,63,79,88,83,70,53,40,31,24,29,38,51,68,76,56,39,26,34,45,58,74,86,89,82,65,77,61,46,59,75,66,49,36,27,22);(3,4,5,7,6,8,9,10)(11,12)(13,17,18,15)(14,16)(20,21,22,24)(23,26,27,33)(25,30,31,29)(28,35,36,46)(32,39,40,42)(34,43,44,38)(37,48,49,63)(41,52,53,59)(45,56,57,55)(47,60,61,51)(50,65,66,78)(54,69,70,79)(58,67,73,75)(62,76,77,72)(64,80,81,68)(71,82,83,87)(74,84,85,88)(86,89,90,91);(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, 1093365);