#27278236

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8880

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1480	C₂
socle	C₂²×C₅₃	#385
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1480	D₅₃
hypocenter	C₅₃	#8880
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1480

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 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)(55,60)(56,57)(59,61)(62,65)(63,70)(64,72)(66,77)(67,74)(68,69)(71,76)(73,75);(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)(55,59)(56,57)(60,61)(62,64,67,70,68,76,66,75)(63,69,71,77,73,65,72,74);(54,55,58,61)(56,59,57,60)(62,63,66,72,68,73,67,71)(64,74,75,65,76,77,70,69)

Polycyclic presentation:

2^8,53|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eh,i^b=i^52,d^c=dh,e^c=eg

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63712);