C₂×#22625176 #34015906

order	18688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#22625176

Splittings

Sylow

C₇₃ ⋊ C₂×#1462

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#64408
commutator	C₂³×C₇₃	C₂³×C₄
Frattini	C₂³	C₂³×#64408
Fitting, prime radical	C₂⁶×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	C₂⁶	#64408
hypocenter	C₇₃	C₂×#1462
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁶

2790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 4 generators on 91 points:

(12,15)(16,18)(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);(3,5)(4,7)(6,10)(8,9);(3,4)(5,9,10,7)(6,8)(11,12,13,16,14,15,17,18)(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

1216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1099490);

C2×#22625176 #34015906