#27304331

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13565

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810410
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₂×C₅₃×#455	C₂
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	C₂×H(5,𝔽₂)×C₅₃	C₄
cosocle	#124×C₅₃	C₂³
hypercenter	C₂×#455	D₅₃
hypocenter	C₅₃	#13565
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#455

496 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 79 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:

(3,13)(7,23)(12,29)(14,31)(15,32)(16,20)(24,28)(26,30)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,14)(5,20)(6,13)(8,12)(9,26)(10,28)(11,23)(15,21)(16,22)(17,32)(18,31)(19,24)(25,29)(27,30);(1,2,4,9,6,11,18,27)(3,12,15,23,13,29,32,7)(5,19,21,25,22,10,17,8)(14,24,20,26,31,28,16,30)(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)

Polycyclic presentation:

2^8,53|a^2=e,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,e^b=efh,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh

Conjugacy classes & character table

860 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89807);