#27430963

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2055 ⋊ C₂
Sylow	C₅₃ ⋊ #25115

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2055	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	#2055	D₅₃
hypocenter	C₅₃	#25115
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2055

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(55,57)(62,66,67,68)(63,70,65,71)(64,74,75,76)(69,73,72,77);(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)(59,61)(62,65,67,63)(64,73,75,77)(66,71,68,70)(69,74,72,76);(55,57)(58,59)(60,61)(62,64,66,74,67,75,68,76)(63,69,71,77,65,72,70,73);(54,55)(56,57)(62,63,68,70,67,65,66,71)(64,72,76,77,75,69,74,73)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 216439);