#27304342

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2126 ⋊ C₂
Sylow	C₅₃ ⋊ #13568

Subgroups

	subgroup	G/H
center	C₄	#16810409
commutator	C₂×Q₈×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₅₃×#2126	C₂
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	#20284063	C₂
cosocle	#124×C₅₃	C₂³
hypercenter	#2126	D₅₃
hypocenter	C₅₃	#13568
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #2126

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

normal subgroups only

This group has 498 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

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

Polycyclic presentation:

2^8,53|a^2=e,c^2=h,d^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,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

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89818);