#27537057

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2916 ⋊ C₂
Sylow	C₅₃ ⋊ #33392

Subgroups

	subgroup	G/H
center	C₂	C₂²×#13133137
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2916	C₂
socle	C₂×C₅₃	C₂²×#102
prime radical	C₂×#16811428	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2916	D₅₃
hypocenter	C₅₃	#33392
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ #2916

1154 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 351 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=f,e^2=g,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,i^b=i^52,d^c=dh

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322533);