#27299941

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#354 ⋊ C₂
Sylow	C₅₃ ⋊ #12768

Subgroups

	subgroup	G/H
center	C₂	#20274661
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#354	C₂
socle	C₂×C₅₃	#1790
prime radical	#20289072	C₂
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₂×#354	D₅₃
hypocenter	C₅₃	#12768
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#354

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

normal subgroups only

This group has 778 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:

(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,15)(11,24)(13,26)(14,27)(18,22)(19,30)(21,29)(25,31)(28,32)(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)(77,78)(79,80)(81,82)(83,84);(1,3,6,14)(2,7,10,21)(4,12,17,27)(5,13,19,28)(8,20,15,29)(9,11,18,25)(16,26,30,32)(22,31,23,24)(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);(1,2)(3,11)(4,15,17,8)(5,18,19,9)(6,10)(7,13,21,28)(12,24,27,31)(14,25)(16,22)(20,26)(23,30)(29,32)

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,i^b=i^52,d^c=dg,f^c=fh,i^c=i^52,e^d=eh,f^d=fh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85417);