#27271859

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7494

Subgroups

	subgroup	G/H
center	C₂	#20272834
commutator	C₂×D₄×C₅₃	C₄²
Frattini	#532	C₂×D₅₃
Fitting	C₅₃×#1536	C₂
socle	C₂×C₅₃	#1390
prime radical, cosocle	C₅₃×#532	C₂²
hypercenter	#1536	D₅₃
hypocenter	C₅₃	#7494
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1536

270 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=dfh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,f^c=fh,f^d=fh

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57335);