C₂×#20276787 #27538467

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20276787

Splittings

Fitting	C₂×C₅₃×#362 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2212

Subgroups

	subgroup	G/H
center	C₂²	#16810817
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#362	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂×#16811061	C₂
hypercenter	C₂×#362	D₅₃
hypocenter	C₅₃	C₂×#2212
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323943);

C2×#20276787 #27538467