C₂×#20272903 #27304552

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272903

Splittings

Fitting	C₂×C₅₃×M₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1418

Subgroups

	subgroup	G/H
center	C₂²	#16810425
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂²×C₅₃	#357
prime radical	C₂×#16811055	C₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	C₂×#1418
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

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

normal subgroups only

This group has 302 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)(58,67)(59,72)(60,75)(61,63)(62,76)(64,71)(65,73)(66,78)(68,81)(69,82)(70,79)(74,77)(80,85)(83,87)(84,86);(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)(57,62,63,78)(58,66)(59,71)(60,73)(64,82,81,86)(65,83,75,80)(67,72,79,77)(68,74)(69,87)(70,76)(84,85);(54,55)

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90028);