#27303517

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13421

Subgroups

	subgroup	G/H
center	C₂²	#16810810
commutator, prime radical, cosocle	C₂²×D₄×C₅₃	C₂³
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₅₃×#2724	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	#2724	D₅₃
hypocenter	C₅₃	#13421
LCS	G ▷ C₂²×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #2724

496 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 77 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)(9,16)(10,17)(12,15)(14,20)(18,19)(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);(1,2)(3,4)(5,7,10,6)(8,13,17,11)(9,12,18,14)(15,16,20,19)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23);(2,3)(5,6,10,7)(8,15,17,20)(9,12,18,14)(11,19,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88993);