#27304361

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1405 ⋊ C₂
Sylow	C₅₃ ⋊ #13573

Subgroups

	subgroup	G/H
center	C₂	AΣL(1,𝔽₁₆)×D₅₃
commutator	C₂×Q₈×C₅₃	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₅₃
Fitting	C₅₃×#1405	C₂
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	2₋¹⁺⁴×D₅₃	C₄
cosocle	C₄⋊Q₈×C₅₃	C₂³
hypercenter	#1405	D₅₃
hypocenter	C₅₃	#13573
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #1405

470 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 59 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)(12,25)(14,17)(15,31)(16,20)(24,30)(28,32)(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);(1,3)(2,7)(4,17)(5,20)(6,13)(8,25)(9,28)(10,30)(11,23)(12,27)(14,19)(15,18)(16,22)(21,31)(24,26)(29,32)(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);(1,2,4,9,6,11,19,29)(3,12,15,23,13,25,31,7)(5,10,21,8,22,26,18,27)(14,30,20,28,17,24,16,32)

Polycyclic presentation:

2^8,53|a^2=e,d^2=h,e^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^c=i^52,e^d=egh,f^d=fh,f^e=fh

Conjugacy classes & character table

470 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89837);