#27287451

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×D₄⋊C₈ ⋊ C₂
Sylow	C₅₃ ⋊ #10408

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×D₄⋊C₈	C₂
socle	C₂³×C₅₃	#120
prime radical	#20279661	C₂
hypercenter	C₂×D₄⋊C₈	D₅₃
hypocenter	C₅₃	#10408
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×D₄⋊C₈

708 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)(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)(54,56,57,55)(59,62)(60,65)(63,64)(67,70)(68,73)(71,72);(54,55)(56,57)(66,67)(68,70)(69,71)(72,73);(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)(55,56)(58,59)(60,62)(61,64)(63,65)(68,73)(70,72)

Polycyclic presentation:

2^8,53|c^2=d,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,i^c=i^52

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72927);