#27271732

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×C₄⋊C₃₂ ⋊ C₂
Sylow	C₅₃ ⋊ #7446

Subgroups

	subgroup	G/H
center	C₂²	#16810425
commutator	C₂×C₁₆×C₅₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₃₂	C₂
socle	C₂²×C₅₃	#357
prime radical	C₂×D₈₄₈	C₄
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#7446
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 2 generators on 93 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)(55,58)(56,60)(62,63)(64,75)(65,81)(66,83)(67,84)(68,70)(69,77)(71,80)(72,76)(73,82)(74,85)(78,88)(79,86)(87,92)(89,91)(90,93);(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)(54,55,56,59,57,58,60,61)(63,69,70,85)(64,74)(65,80)(66,76)(67,82)(71,88,72,89)(73,90,84,87)(75,81,86,83)(77,79)(78,93)(91,92)

Polycyclic presentation:

2^8,53|a^2=d,c^2=fg,d^2=efgh,e^2=gh,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,i^a=i^52,c^b=cf,d^b=defg,e^b=eg,f^b=fg,g^b=gh,i^b=i^52

Conjugacy classes & character table

2129 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57208);