#27557462

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×D₄×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #35028

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₄²×D₄×C₅₃	C₂
prime radical	#20279671	C₂
hypercenter	C₄²×D₄	D₅₃
hypocenter	C₅₃	#35028
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄

1816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(55,59)(58,61)(62,65,66,64)(63,69,67,68)(70,73,74,72)(71,75,77,76);(56,60)(58,61)(62,64,66,65)(63,68,67,69)(70,72,74,73)(71,76,77,75);(56,60)(58,61)(70,72,74,73)(71,75,77,76);(54,56)(55,58)(57,60)(59,61)(63,67)(68,69)(70,72,74,73)(71,75,77,76);(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)(54,55)(56,58)(57,59)(60,61)(62,63)(64,68)(65,69)(66,67)(70,71)(72,76)(73,75)(74,77)

Polycyclic presentation:

2^8,53|b^2=g,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=eh,i^a=i^52,c^b=ch,d^b=dh

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 342938);