#27304346

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#354 ⋊ C₂
Sylow	C₅₃ ⋊ #13569

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810409
commutator	C₂×Q₈×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₂×C₅₃×#354	C₂
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	C₂×#16810795	C₂
cosocle	#124×C₅₃	C₂³
hypercenter	C₂×#354	D₅₃
hypocenter	C₅₃	#13569
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#354

492 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 71 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:

(3,12)(7,23)(11,26)(13,17)(14,15)(24,25)(29,30)(31,32)(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)(4,17)(5,14)(6,12)(8,26)(9,24)(10,23)(11,27)(13,18)(15,22)(16,30)(19,31)(20,25)(21,32)(28,29)(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)(3,11,14,29)(4,16,18,28)(5,20,22,9)(6,10)(7,13,24,31)(8,19)(12,26,15,30)(17,25,32,23)(21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89822);