#27539005

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33659

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810939
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3060	C₂
socle	C₂×C₅₃	C₂×#481
hypercenter	#3060	D₅₃
hypocenter	C₅₃	#33659
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3060

546 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 167 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)(54,56,59,64)(55,61,62,75)(57,65,68,77)(58,66,70,78)(60,73,72,83)(63,76,71,81)(67,79,82,85)(69,80,74,84);(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)(54,55,58,71,59,62,70,63)(56,61,66,81,64,75,78,76)(57,69,67,72,68,74,82,60)(65,80,79,83,77,84,85,73);(56,64)(57,68)(60,72)(61,75)(66,78)(67,82)(69,74)(76,81);(55,60)(57,67)(58,70)(61,73)(62,72)(63,69)(65,79)(66,78)(68,82)(71,74)(75,83)(76,80)(77,85)(81,84)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324481);