#27281546

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1297 ⋊ C₂
Sylow	C₅₃ ⋊ #9411

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1297	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂⁴×D₅₃	D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1297	D₅₃
hypocenter	C₅₃	#9411
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1297

606 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 93 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 — 3 generators on 85 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)(54,57)(55,60)(56,62)(58,66)(59,67)(61,64)(63,65)(68,69)(72,80)(76,84)(79,85)(81,82);(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)(54,56)(55,59,61,68)(57,65)(58,63)(60,69,64,67)(62,66)(70,72)(71,76,78,84)(73,81)(74,82)(75,80)(77,79,83,85);(54,55)(56,59,63,68)(57,64,66,60)(58,61)(62,67)(65,69)(70,71,74,77,75,78,73,83)(72,79,82,84,80,85,81,76)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,i^b=i^52,d^c=dh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 67022);