#27303547

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13425

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, prime radical, cosocle	C₂²×D₄×C₅₃	C₂³
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#352	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	C₂×#352	D₅₃
hypocenter	C₅₃	#13425
LCS	G ▷ C₂²×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

508 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 73 points:

(1,3,4,2)(5,8)(6,11)(7,13)(9,18)(10,17)(12,20)(14,15)(16,19)(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)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72);(1,2)(3,4)(5,7,10,6)(8,13,17,11)(9,12,19,14)(15,16,20,18);(2,3)(5,6,10,7)(8,15,17,20)(9,14,19,12)(11,18,13,16)(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,c^2=d,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,i^a=i^52,c^b=cf,e^b=eg,e^c=eh,g^c=gh,i^c=i^52,f^e=fh

Conjugacy classes & character table

707 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89023);