#27280506

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9254

Subgroups

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#460	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#127×C₅₃	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#460	D₅₃
hypocenter	C₅₃	#9254
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#460

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3392

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

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dh,e^b=egh,f^b=fh,i^b=i^52,d^c=dgh,e^c=egh,f^c=fgh

Conjugacy classes & character table

1031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 65982);