#27280327

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1802 ⋊ C₂
Sylow	C₅₃ ⋊ #9229

Subgroups

	subgroup	G/H
center	C₂³	#13133189
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1802	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂×#16810818	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1802	D₅₃
hypocenter	C₅₃	#9229
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1802

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

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, 65803);