#27300409

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1386 ⋊ C₂
Sylow	C₅₃ ⋊ #12873

Subgroups

	subgroup	G/H
center	C₂	#20274800
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1386	C₂
socle	C₂×C₅₃	#1823
prime radical	C₂×#11719945	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1386	D₅₃
hypocenter	C₅₃	#12873
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

380 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 103 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)(54,56)(55,61,62,78)(57,70)(58,74)(59,67)(60,69,65,82)(63,81,80,77)(64,76,73,79)(66,71)(68,75)(72,84)(83,85);(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)(54,56)(55,60)(57,66)(58,68)(59,67)(61,69)(62,65)(63,76)(64,77)(70,71)(72,83)(73,81)(74,75)(78,82)(79,80)(84,85);(54,55)(56,65,67,60)(57,69)(58,73)(59,62)(61,66,78,83)(63,71)(64,75)(68,77,74,81)(70,79,84,76)(72,82)(80,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=d,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,f^b=fh,i^b=i^52,e^c=eg,f^c=fh,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, 85885);