#27415082

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23847

Subgroups

	subgroup	G/H
center	C₄	C₅₃⋊#401
commutator, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂×C₅₃	C₂×#401
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#23847
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #1740

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,59,71)(55,62,63,77)(56,66,67,80)(57,69,70,82)(60,73,64,78)(61,75,76,85)(65,79,68,81)(72,83,74,84);(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)(54,57)(55,61,63,76)(56,65)(58,69)(59,70)(60,72,64,74)(62,75,77,85)(66,79)(67,68)(71,82)(73,83,78,84)(80,81);(54,56)(55,60)(57,68,70,65)(58,66)(59,67)(61,74,76,72)(62,73)(63,64)(69,81,82,79)(71,80)(75,84,85,83)(77,78);(54,55)(56,64,67,60)(57,61,70,76)(58,62)(59,63)(65,74)(66,78,80,73)(68,72)(69,75,82,85)(71,77)(79,84)(81,83)

Polycyclic presentation:

2^8,53|a^2=efg,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 200558);