#27280620

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#332 ⋊ C₂
Sylow	C₅₃ ⋊ #9271

Subgroups

	subgroup	G/H
center	C₂²	#16810499
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#332	C₂
socle	C₂²×C₅₃	#386
prime radical	#16811173	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#332	D₅₃
hypocenter	C₅₃	#9271
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#332

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 points:

(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)(55,60)(56,57)(59,61)(62,64)(63,68)(65,71)(66,72)(67,74)(69,77)(70,73)(75,76);(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)(55,59)(56,57)(60,61)(62,63,65,67,66,69,73,75)(64,68,71,74,72,77,70,76);(54,55,56,60,58,61,57,59)(63,67)(64,70)(65,73)(69,75)(71,72)(74,76)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfh,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh,i^c=i^52

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, 66096);