#27271452

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1270 ⋊ C₂
Sylow	C₅₃ ⋊ #7344

Subgroups

	subgroup	G/H
center	C₂²	#16810411
commutator	C₂×D₄×C₅₃	C₂×C₈
Frattini	#390	C₂×D₅₃
Fitting	C₅₃×#1270	C₂
socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
prime radical	C₂×#13133262	C₄
cosocle	C₅₃×#390	C₂²
hypercenter	#1270	D₅₃
hypocenter	C₅₃	#7344
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #390 ◁ #1270

376 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 77 points:

(9,11)(10,14)(12,21)(13,20)(15,18)(16,24)(17,23)(19,22)(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)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76);(1,2,3,5,4,6,7,8)(9,10)(11,18,19,14)(12,16,22,15)(13,17)(20,24,21,23)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)

Polycyclic presentation:

2^8,53|a^2=d,d^2=f,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,d^b=deh,e^b=eh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eh,e^d=eh

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56928);