C₂×#20272922 #27304610

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#20272922

Splittings

Sylow

C₅₃ ⋊ C₂×#1426

Subgroups

	subgroup	G/H
center	C₂²	#16810450
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	#367
prime radical	C₂³×C₅₃	SD₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	C₂×#1426
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#368

190 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(56,57,63,79,59,65,58,62,72,83,67,76,68,74,78,85,61,64,81,87,73,82,70,75,60,66,80,86,71,84,69,77);(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)(57,62,64,75)(58,67)(59,71)(60,72)(63,78)(65,79,82,87)(66,85,83,77)(68,73)(69,81)(70,80)(74,86,84,76);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90086);